Loading [MathJax]/jax/element/mml/optable/MathOperators.js
Research article

Is Alzheimers Disease Infectious?
Relative to the CJD Bacterial Infection Model of Neurodegeneration

  • Alzheimer's disease (AD) has been recently considered as a possible brain infection related to the Creutzfeldt-Jakob disease (CJD) transmissible dementia model. As with CJD, there is controversy whether the infectious agent is an amyloid protein (prion theory) or a bacterium. In this review, we show that the prion theory lacks credibility because spiroplasma, a tiny wall-less bacterium, is clearly involved in the pathogenesis of CJD and the prion amyloid can be separated from infectivity. In addition to prion amyloid deposits, the transmissible agent of CJD is associated with amyloids (A-β, Tau, and α-synuclein) characteristic of other neurodegenerative diseases including AD and Parkinsonism. Reports of spiroplasma inducing formation of α-synuclein in tissue culture and Borrelia spirochetes inducing formation of A-β and Tau in tissue culture suggests that bacteria may have a role in the pathogenesis of the neurodegenerative diseases.

    Citation: Frank O. Bastian. Is Alzheimers Disease Infectious?Relative to the CJD Bacterial Infection Model of Neurodegeneration[J]. AIMS Neuroscience, 2015, 2(4): 240-258. doi: 10.3934/Neuroscience.2015.4.240

    Related Papers:

    [1] Neda Askari, Hassan Momtaz, Elahe Tajbakhsh . Acinetobacter baumannii in sheep, goat, and camel raw meat: virulence and antibiotic resistance pattern. AIMS Microbiology, 2019, 5(3): 272-284. doi: 10.3934/microbiol.2019.3.272
    [2] Aynur Aybey, Elif Demirkan . Inhibition of Pseudomonas aeruginosa biofilm formation and motilities by human serum paraoxonase (hPON1). AIMS Microbiology, 2016, 2(4): 388-401. doi: 10.3934/microbiol.2016.4.388
    [3] Liang Wang, Zuobin Zhu, Huimin Qian, Ying Li, Ying Chen, Ping Ma, Bing Gu . Comparative genome analysis of 15 clinical Shigella flexneri strains regarding virulence and antibiotic resistance. AIMS Microbiology, 2019, 5(3): 205-222. doi: 10.3934/microbiol.2019.3.205
    [4] Maureen U. Okwu, Mitsan Olley, Augustine O. Akpoka, Osazee E. Izevbuwa . Methicillin-resistant Staphylococcus aureus (MRSA) and anti-MRSA activities of extracts of some medicinal plants: A brief review. AIMS Microbiology, 2019, 5(2): 117-137. doi: 10.3934/microbiol.2019.2.117
    [5] Kamila Tomoko Yuyama, Thaís Souto Paula da Costa Neves, Marina Torquato Memória, Iago Toledo Tartuci, Wolf-Rainer Abraham . Aurantiogliocladin inhibits biofilm formation at subtoxic concentrations. AIMS Microbiology, 2017, 3(1): 50-60. doi: 10.3934/microbiol.2017.1.50
    [6] Joseph O. Falkinham . Mycobacterium avium complex: Adherence as a way of life. AIMS Microbiology, 2018, 4(3): 428-438. doi: 10.3934/microbiol.2018.3.428
    [7] Itziar Chapartegui-González, María Lázaro-Díez, Santiago Redondo-Salvo, Elena Amaro-Prellezo, Estefanía Esteban-Rodríguez, José Ramos-Vivas . Biofilm formation in Hafnia alvei HUMV-5920, a human isolate. AIMS Microbiology, 2016, 2(4): 412-421. doi: 10.3934/microbiol.2016.4.412
    [8] Tatyana V. Polyudova, Daria V. Eroshenko, Vladimir P. Korobov . Plasma, serum, albumin, and divalent metal ions inhibit the adhesion and the biofilm formation of Cutibacterium (Propionibacterium) acnes. AIMS Microbiology, 2018, 4(1): 165-172. doi: 10.3934/microbiol.2018.1.165
    [9] Yunfeng Xu, Attila Nagy, Gary R. Bauchan, Xiaodong Xia, Xiangwu Nou . Enhanced biofilm formation in dual-species culture of Listeria monocytogenes and Ralstonia insidiosa. AIMS Microbiology, 2017, 3(4): 774-783. doi: 10.3934/microbiol.2017.4.774
    [10] Stephen H. Kasper, Ryan Hart, Magnus Bergkvist, Rabi A. Musah, Nathaniel C. Cady . Zein nanocapsules as a tool for surface passivation, drug delivery and biofilm prevention. AIMS Microbiology, 2016, 2(4): 422-433. doi: 10.3934/microbiol.2016.4.422
  • Alzheimer's disease (AD) has been recently considered as a possible brain infection related to the Creutzfeldt-Jakob disease (CJD) transmissible dementia model. As with CJD, there is controversy whether the infectious agent is an amyloid protein (prion theory) or a bacterium. In this review, we show that the prion theory lacks credibility because spiroplasma, a tiny wall-less bacterium, is clearly involved in the pathogenesis of CJD and the prion amyloid can be separated from infectivity. In addition to prion amyloid deposits, the transmissible agent of CJD is associated with amyloids (A-β, Tau, and α-synuclein) characteristic of other neurodegenerative diseases including AD and Parkinsonism. Reports of spiroplasma inducing formation of α-synuclein in tissue culture and Borrelia spirochetes inducing formation of A-β and Tau in tissue culture suggests that bacteria may have a role in the pathogenesis of the neurodegenerative diseases.


    The present work is motivated by the study of nonlocal peridynamics models initially proposed by Silling in [31]. In particular, the state-based peridynamics model given in [31,32,33] postulates that the total strain energy for constitutively linear, isotropic solid undergoing deformation is given by

    $ Wρ(u)=βΩ(Dρ(u)(x))2dx+αΩΩρ(xx)(D(u)(x,x)1dDρ(u)(x))2dxdx $ (1.1)

    where $ \Omega\subset \mathbb{R}^d $ is a bounded domain occupied by the solid material, the kernel $ \rho(|{ \boldsymbol \xi}|) $ is a nonnegative locally integrable and radial weight function that measures the interaction strength between material particles at position $ {\bf x} $ and $ {\bf x}' $, $ {\bf u} $ is a displacement field, $ {\mathscr D}({\bf u}) $ is a rescaled nonlocal operator on $ {\bf u} $ defined by [11]

    $ D(u)(x,x)=(u(x)u(x))|xx|(xx)|xx|=(u(x)u(x))T(xx)|xx|2, $ (1.2)

    representing a (unit-less) linearized nonlocal strain [32] and the operator $ \mathfrak{D}_{\rho} $ is a nonlocal linear operator (a weighted version of $ {\mathscr D} $ [11,12]), called 'nonlocal divergence', which is defined as

    $ Dρ(u)(x):=p.v.Ωρ(xx)D(u)(x,x)dx $ (1.3)

    which is a means of incorporating the effect of the collective deformation of a neighborhood of $ {\bf x} $ into the model. The positive constants $ \alpha $ and $ \beta $ are proportional to the shear and bulk moduli of the material, respectively. The quadratic energy in (1.1) is a generalization of the bond-based model that was introduced in [31] and studied in [1,13,15,23,37] that takes in to account the linearized strain due to the dilatation and the deviatoric portions of the deformation. Mathematical analysis of linearized peridynamic models have been extensively studied in [10,11,12,13,15,23,24,37] along with results geared towards nonlinear models in [3,4,9,14,19,25].

    For $ \rho \in L^{1}_{loc} $, it is not difficult to show (see Lemma 2.3 below) that the energy space associated with the energy functional $ W_\rho $, $ \{{\bf u}\in L^{2}(\Omega; \mathbb{R}^{d}): W_{\rho}({\bf u}) < \infty \} $, is precisely

    $ Sρ,2(Ω)={uL2(Ω;Rd):|u|2Sρ,2<}, $ (1.4)

    where the seminorm $ |{\bf u}|_{\mathcal{S}_{\rho, 2}} $ is defined by

    $ |{\bf u}|^{2}_{\mathcal{S}_{\rho, 2}} : = \int_{\Omega } \int_{\Omega } \rho( {\bf y} - {\bf x})\left|\frac{({\bf u}( {\bf y}) - {\bf u} ({ {\bf x}}))}{| {\bf y}- {\bf x}|} \cdot \frac{( {\bf y} - {\bf x})}{| {\bf y} - {\bf x}|}\right|^{2}d {\bf y} d {\bf x}. $

    Notice that $ |{\bf u}|_{\mathcal{S}_{\rho, 2}} = 0, $ if and only if $ {\bf u} $ is an infinitesimal rigid vector field. We denote the class of infinitesimal rigid displacements by

    $ \mathcal{R}: = \{{\bf u}: {\bf u}({\bf x}) = \mathbb{B} {\bf x} + {\bf v}, \mathbb{B}^{T} = -\mathbb{B}, {\bf v} \in \mathbb{R}^{d}\}. $

    It has been shown in [23,25] that $ \mathcal{S}_{\rho, 2}(\Omega) $ with the natural norm

    $ \|{\bf u}\|_{\mathcal{S}_{\rho, 2}} = {(\|{\bf u}\|^2_{L^2} + |{\bf u}|^2_{\mathcal{S}_{\rho, 2}})^{1/2}} $

    is a separable Hilbert space. In the event that $ \rho({ \boldsymbol \xi})|{ \boldsymbol \xi}|^{-2} \in L^{1}_{loc}(\mathbb{R}^{d}) $, then the space $ \mathcal{S}_{\rho, 2}(\Omega) $ coincides with $ L^{2}(\Omega, \mathbb{R}^{d}) $. Otherwise, $ \mathcal{S}_{\rho, 2}(\Omega) $ is a proper subset of $ L^{2}(\Omega, \mathbb{R}^{d}) $ that is, nevertheless, big enough to contain $ W^{1, 2}(\Omega; \mathbb{R}^{d}) $ and there exists a constant $ C = C(d, 2, \Omega) > 0 $ such that

    $ |{\bf u}|^{2}_{\mathcal{S}_{\rho, 2}} \leq C \|\text{Sym}(\nabla {\bf u})\|^{2}_{L^{2}} \|\rho\|_{L^{1}(\mathbb{R})}, \quad\quad\forall {\bf u}\in W^{1, 2}(\Omega; \mathbb{R}^{d}) $

    where $ \text{Sym}(\nabla {\bf u}) = \frac{1}{2} (\nabla {\bf u} + \nabla {\bf u} ^{T}) $ is the symmetric part of the gradient $ \nabla {\bf u} $.

    Under the additional assumption that $ \rho $ is positive in a small neighborhood of the origin, it is shown in [23, Theorem 1], via an application of Lax-Milgram, that for any applied load $ {\bf f}\in L^{2}(\Omega; \mathbb{R}^{d}) $, the potential energy

    $ Eρ(u)=Wρ(u)Ωfudx $ (1.5)

    has a minimizer over any weakly closed subset $ V $ of $ \mathcal{S}_{\rho, 2}(\Omega) $ such that $ V\cap \mathcal{R} = \{{ \bf 0}\} $. See also [25, Theorem 1.1] for the more general convex energies of $ p $-growth.

    The analysis of the convergence of variational problems of the type in (1.5) associated with a sequence of parameterized kernels has garnered a lot of attention in recent years. Namely, if we have a sequence of locally integrable radial kernels $ \rho_{n} $, how do the associated potential energies $ {E_{\rho_n}} $, as well as their minimizers behave as $ n\to \infty $? Clearly, this will depend first on the behavior of the convergence properties of the sequence of kernels. In fact, it is shown in [25] that if the sequence of $ L^{1} $ kernels $ \{\rho_n\} $ converge in the sense of measures to a measure with atomic mass at $ 0 $ (Dirac-measure at $ 0 $) and for each $ n $, $ r^{-2}\rho_{n}(r) $ is nonincreasing, then the sequence $ \{E_{\rho_n}\} $ variationally converges to the classical Navier-Lamé potential energy $ E_0 $ given by

    $ E_{0}({\bf u}) = \mu \int_{\Omega}|\text{Sym}(\nabla {\bf u})|^2 dx + \frac{\lambda}{2}\int_{\Omega} (\text{div} ({\bf u}))^2 dx - \int_{\Omega} {\bf f}\cdot {\bf u} dx, $

    where $ \mu $ and $ \lambda $ are constants that can be expressed in terms of $ \alpha $ and $ \beta $. This is what is called nonlocal-to-local convergence and the result is used as a rigorous justification that state-based peridynamics modeling recovers the classical linearized elasticity models in the event of vanishing nonlocality.

    In this paper, we consider another type of convergence of sequence of kernels and study the behavior of the associated energy functionals, which leads to nonlocal-to-nonlocal convergence. More specifically, suppose we are given a nonnegative kernel $ \rho\in L^{1}_{loc}(\mathbb{R}^d) $ with the property that

    $ ρ is radial ,ρ(ξ)>0 for ξ is close to 0,|ξ|2ρ(|ξ|) is nonincreasing in |ξ|, $ (1.6)

    and

    $ limδ0δ2(Bδρ(ξ)dξ)1=0. $ (1.7)

    and consider a sequence of nonnegative, radial kernels $ \{\rho_{n}\}_{n\in \mathbb{N}} $ each satisfying (1.6) and that

    $ ρnρa.e. and ρnρa.e. in Rd. $ (1.8)

    It then follows that $ \rho_n \to \rho $ strongly in $ L^{1}_{loc}(\mathbb{R}^d) $ as $ n\to \infty $. We will establish a clear connection between the sequence of energies $ \{E_{\rho_n}\} $ and $ E_{\rho} $. Most importantly, we will show that minimizers of the energies $ E_{\rho_n} $ over an admissible class will converge to a minimizer of $ E_{\rho} $ over the same admissible class. The notion of variational convergence we use is $ \Gamma $-convergence (see [21]) which we define below. The advantage of $ \Gamma $-convergence is that under the additional assumption of equicoercivity of the functionals it implies the convergence of minimizers as well [21, Theorem 7.8 and Corollary 7.202].

    Definition 1.1. Suppose that $ \overline{E}_{n}: L^{2}(\Omega; \mathbb{R}^{d})\to \mathbb{R}\cup \{\infty\} $, $ \forall 1\leq n\leq \infty $. We say that the sequence $ \overline{E}_{n} $ $ \Gamma $- converges to $ \overline{E}_{\infty} $ in the $ L^{2} $-topology if and only if

    a) for every sequence $ \{{\bf u}_{n}\}\in L^{2}(\Omega; \mathbb{R}^{d}) $ with $ {\bf u}_{n}\to {\bf u} $ in $ L^{2}(\Omega; \mathbb{R}^{d}), $ as $ n\to\infty, $ we have $ \overline{E}_{\infty}({\bf u})\leq \liminf_{n\to \infty}\overline{E}_{n}({\bf u}_{n}) $,

    b) and for every $ {\bf u}\in L^{2}(\Omega; \mathbb{R}^{d}) $ there exists a recovery sequence $ {\bf u}_{n}\to {\bf u} $ in $ L^{2}(\Omega; \mathbb{R}^{d}), $ such that $ \overline{E}_{\infty}({\bf u}) = \lim_{n\to \infty}\overline{E}_{n}({\bf u}_{n}). $

    The following is one of the main results of the paper on the variational limit of the nonlocal functionals $ \{\overline{E}_{n}\} $.

    Theorem 1.2. Suppose $ \rho $ and $ \{\rho_n\} $ satisfies (1.6)–(1.8). The sequence of functionals $ \overline{E}_{n} $ $ \Gamma- $converges in the strong $ L^{2}(\Omega; \mathbb{R}^{d}) $ topology to the functional $ \overline{E}_{\rho} $, where the extended functionals $ \{\overline{E}_{n}({\bf u})\}_{n\leq \infty} $ are defined as

    $ ¯En(u)={Eρn(u),if uSρn,2(Ω),,if uL2(Ω;Rd)Sρn,2(Ω), $ (1.9)

    where $ \mathcal{S}_{\rho_n, 2}(\Omega) $ and $ \{E_{\rho_{n}}\}_{n < \infty} $ are defined as before in (1.4) and (1.5), respectively, where $ \rho $ is replaced by $ \rho_n $. The extended functional $ \overline{E}_{\rho} $ is similarly defined. Moreover, if $ V $ is a weakly closed subset of $ L^{2}(\Omega; \mathbb{R}^{d}) $ such that $ V\cap \mathcal{R} = \{ { \bf 0}\} $, and for each n, $ {\bf u}_{n} $ minimizes $ E_{\rho_n} $ over $ V\cap \mathcal{S}_{\rho_n, 2}(\Omega) $, then the sequence $ \{{\bf u}_n\} $ is precompact in $ L^{2}(\Omega; \mathbb{R}^{d}) $ with any limit point belonging to $ \mathcal{S}_{\rho, 2}(\Omega) $ and minimizing $ E_{\rho} $ over $ V\cap \mathcal{S}_{\rho, 2}(\Omega) $.

    Although the discussion above is focused on the case of quadratic peridynamic energies for ease of explaining the main ideas, the result can naturally be extended to small strain nonlocal nonlinear peridynamic models with $ p $-growth, for $ p\geq 2 $, that have been introduced in [33] and whose variational analysis investigated in [25].

    We will prove Theorem 1.2 in the sections that follow. But we would like to highlight that this result has an important implication in the numerical approximation of minimizers of $ E_{\rho} $ over an admissible class. Indeed, compactness results have been quite useful for analyzing numerical approximations of nonlocal problems in various contexts such as [8,34,35]. In the context discussed in this work, let us take for an example that $ \rho({ \boldsymbol \xi}) = \frac{1}{|{ \boldsymbol \xi}|^{d + 2(s-1)}} $ for $ s\in (0, 1) $. This kernel satisfies (1.6) and (1.7). It is also clear that $ \rho({ \boldsymbol \xi})|{ \boldsymbol \xi}|^{-2} $ is not integrable on any bounded domain containing $ { \bf 0} $. In the event $ \Omega $ has a smooth boundary, the energy space $ \mathcal{S}_{\rho, 2}(\Omega) $ coincides with the fractional Sobolev space $ H^{s}(\Omega; \mathbb{R}^d) $ see [27,30]. In particular, if $ {1/2} < s < 1 $, then all functions in $ \mathcal{S}_{\rho, 2}(\Omega) $ have continuous representative. Now, if $ V\subset\mathcal{S}_{\rho, 2}(\Omega) $ is a weakly closed subset of $ L^{2}(\Omega; \mathbb{R}^{d}) $ such that $ V\cap \mathcal{R} = \{ { \bf 0}\} $, a minimizer of $ E_{\rho} $ over $ V $ exists (and will be in $ H^{s}(\Omega; \mathbb{R^d}) $). The analysis of the existence and uniqueness of the minimizer $ {\bf u} $ of this quadratic energy can also, equivalently, be found by solving the corresponding Euler-Lagrange equation. The latter gives us a way of numerically solving the solution by writing it first in the weak form and then applying the Galerkin approach of choosing a finite-dimensional subspace $ \mathcal{M} \subset V $ to solve for a projected solution of $ {\bf u} $ on $ \mathcal{M} $. Notice that for $ s\in (0, 1/2) $ the finite-dimensional subspace $ \mathcal{M} $ can contain discontinuous functions, while for $ s\in (1/2, 1) $, all the elements of $ \mathcal{M} $ must be continuous in order for $ \mathcal{M} $ to be conforming, that is, for $ \mathcal{M}\subset \mathcal{S}_{\rho, 2}(\Omega) $. In the latter case, if one wants to employ the advantageous discontinuous Galerkin approximation, which is now nonconforming, one needs to find an effective way to implement it to the model problem. The result in Theorem 1.2 will allow us to develop approximation schemes by solving a sequence of Euler-Lagrange equations of modified energies. To demonstrate this, define the sequence of kernels

    $ \rho_{n}({ \boldsymbol \xi}) = \left\{ρ(ξ)if ρ(ξ)n|ξ|2n|ξ|2if ρ(ξ)n|ξ|2 \right. . $

    It is not difficult to check that, for each $ n $, $ \rho_{n} $ satisfies (1.6), (1.8), and that the functions $ \rho_{n}({ \boldsymbol \xi})|{ \boldsymbol \xi}|^{-2} \in L^{1}_{loc}(\mathbb{R}^{n}) $ are just truncations of the fractional kernel $ |{ \boldsymbol \xi}|^{-d-2s} $ (at level $ n $). As discussed before, the energy space associated with $ E_{\rho_n} $ will coincide with $ L^{2}(\Omega; \mathbb{R}^{d}) $ and a unique minimizer $ {\bf u}_n \in V\cap \mathcal{S}_{n} $ of $ E_{\rho_n} $ exists. Since the admissible space is a subspace of $ L^{2}(\Omega; \mathbb{R}^{d}) $ that avoids nontrivial infinitesimal rigid displacements, we may use discontinuous finite element spaces, denoted by $ \mathcal{M}_{n, h} $, for the standard conforming Galerkin approximation to the solution of the Euler-Lagrange equation associated with the energy $ E_{\rho_n}. $ This, in turn, can be viewed as a nonconforming discontinuous Galerkin scheme to numerically solve the original Euler-Lagrange equations when the discretization parameter $ h $ goes to zero and at the same time the truncation level $ n $ goes to infinity. Intuitively, for large $ n $, by Theorem 1.2, $ {\bf u} $ is approximated by $ {\bf u}_n $ (in the $ L^{2} $ norm), and then $ {\bf u}_n $ will be numerically approximated by $ {\bf u}_{n, h}\in \mathcal{M}_{n, h} $. The proper convergence analysis of this nonconforming numerical scheme as $ h\to 0 $ and $ n\to \infty $ simultaneously has been carried out in [34] in the special case of scalar nonlocal problems when the subspace $ V $ is the set of scalar-valued functions $ {\bf u}\in L^{2}(\Omega; \mathbb{R}^{d}) $ such that $ {\bf u} $ vanishes outside of a fixed set $ \Omega' $ which is compactly contained in $ \Omega $. The analysis in [34] makes use of the framework of asymptotically compatible schemes for parameter-dependent problems first developed in [35] and the vanishing of the functions in the admissible class around the volumetric-boundary $ \Omega\setminus \Omega' $ is crucial for employing certain compactness arguments. To extend the convergence analysis in [34] to the case of a system of strongly coupled nonlocal equations, the variations problems associated with (1.5), solved over any admissible set that does not include infinitesimal rigid vector fields, Theorem 1.2 as well as some of the compactness results we prove in this paper will be crucial. Such analysis on nonconforming discontinuous Galerkin numerical schemes to systems of nonlocal equations under discussion will be carried out in a future work.

    Although it is beyond the scope of this work, in passing, we would like to note that this way of developing a nonconforming numerical scheme is also applicable to fractional PDEs where singular kernels are more common [5,36]. The idea is the same where we use less singular kernels with truncation both at origin and at infinity to do approximation of fractional PDEs. In this case, sequential compactness of nonlocal spaces associated with the truncated fractional kernel together with the compact embedding of fractional Sobolev spaces in $ L^{p} $ can be used to carry out the analysis of the resulting asymptotically compatible schemes [35,36].

    The proof of Theorem 1.2 fundamentally depends on some structural properties of the nonlocal space $ \mathcal{S}_{\rho, 2}(\Omega) $, chief among them are compact embedding into $ L^{2}(\Omega; \mathbb{R}^d) $ and a Poincaré-type inequality, which we will establish in this paper. In fact, these properties remain true even for the spaces $ \mathcal{S}_{\rho, p}(\Omega) $, where for $ 1\leq p < \infty $,

    $ \mathcal{S}_{\rho, 2}(\Omega) = \{{\bf u}\in L^{p}(\Omega;\mathbb{R}^d): |{\bf u}|_{\mathcal{S}_{\rho, p}} < \infty\} {, } $

    and

    $ |{\bf u}|_{\mathcal{S}_{\rho, p}} ^{p} = \int_{\Omega } \int_{\Omega } \rho( {\bf y} - {\bf x})\left|\frac{({\bf u}( {\bf y}) - {\bf u} ({ {\bf x}}))}{| {\bf y}- {\bf x}|} \cdot \frac{( {\bf y} - {\bf x})}{| {\bf y} - {\bf x}|}\right|^{p}d {\bf y} d {\bf x} $

    gives a semi norm. It is shown in [20,25] that, for any $ 1\leq p < \infty $, $ \mathcal{S}_{\rho, p}(\Omega) $ is a separable Banach space with the norm

    $ \|{\bf u}\|_{\mathcal{S}_{\rho, p}} = \left(\|{\bf u}\|_{L^{p}}^{p} + |{\bf u}|^{p}_{\mathcal{S}_{\rho, p}}\right)^{1/p}, $

    and is reflexive if $ 1 < p < \infty $ and a Hilbert space for $ p = 2 $. If $ |{ \boldsymbol \xi}|^{-p}\rho({ \boldsymbol \xi}) \in L_{loc}^{1}(\mathbb{R}^{d}) $, then a simple calculation shows that $ \mathcal{S}_{\rho, p}(\Omega) = L^{p}(\Omega; \mathbb{R}^{d}) $. On the other hand, in the case where $ |{ \boldsymbol \xi}|^{-p}\rho({ \boldsymbol \xi}) \notin L^{1}_{loc}(\mathbb{R}^{d}) $, $ \mathcal{S}_{\rho, p}(\Omega) $ is a proper subset of $ L^{p}(\Omega; \mathbb{R}^{d}) $. Under some extra assumptions on the kernel $ \rho $, the space is known to support a Poincaré-Korn type inequality over subsets that have trivial intersections with $ \mathcal{R} $. These functional analytic properties of the nonlocal space can be used to demonstrate the well-posedness of some nonlocal variational problems using the direct method of calculus of variations, see [25] for more discussions.

    As in the case of $ p = 2 $, we assume that for a given $ 1\leq p < \infty, $

    $ ρ is radial,ρ(r)>0 for r is close to 0, and rpρ(r) is nonincreasing in r, $ (1.10)

    and

    $ limδ0δp(Bδ(0)ρ(ξ)dξ)1=0. $ (1.11)

    We now state the compactness result whose proof is one of the main objectives of the present work.

    Theorem 1.3 ($ L^{p} $ compactness). Let $ 1\leq p < \infty $ and let $ \rho \in L^{1}_{loc}(\mathbb{R}^{d}) $ be nonnegative and satisfying (1.10) and (1.11). Suppose also that $ \Omega\subset \mathbb{R}^{d} $ is a domain with Lipschitz boundary. Then $ \mathcal{S}_{\rho, p}(\Omega) $ is compactly embedded in $ L^{p}(\Omega; \mathbb{R}^{d}) $. That is, any bounded sequence $ \{{\bf u}_{n}\} $ in $ \mathcal{S}_{\rho, p}(\Omega) $ is precompact in $ L^{p}(\Omega; \mathbb{R}^{d}) $. Moreover, any limit point is in $ \mathcal{S}_{\rho, p}(\Omega) $.

    The condition given by (1.11) requires $ \rho $ to have an adequate singularity near $ { \bf 0} $. A straightforward calculation shows that the kernels satisfying (1.11) include $ \rho({ \boldsymbol \xi}) = |{ \boldsymbol \xi}|^{-(d + p(s-1))} $, for any $ p\in [1, \infty) $, and any $ s\in (0, 1) $, and $ \rho({ \boldsymbol \xi}) = -|{ \boldsymbol \xi}|^{p-d} \ln(|{ \boldsymbol \xi}|) $. It is no surprise that (1.11) is violated if $ |{ \boldsymbol \xi}|^{-p}\rho({ \boldsymbol \xi}) $ is a locally integrable function (and therefore, $ \mathcal{S}_{\rho, p}(\Omega) = L^{p}(\Omega; \mathbb{R}^{d}) $), and in fact, in this case

    $ \liminf\limits_{\delta \to 0}{\delta^{p} }\left({ \int_{B_{\delta}({ \bf 0}) } \rho({ \boldsymbol \xi})d{ \boldsymbol \xi} }\right)^{-1} = \infty, $

    see [25]. It is not clear whether condition (1.11) is necessary for compact embedding even for the class of kernels that are radial and nonincreasing. There are radial kernels with the property that $ |{ \boldsymbol \xi}|^{-p}\rho({ \boldsymbol \xi}) $ is (locally) nonintegrable, and

    $ \lim\limits_{\delta \to 0}{\delta^{p} }\left({ \int_{B_{\delta} ({ \bf 0})} \rho({ \boldsymbol \xi})d{ \boldsymbol \xi} } \right)^{-1} = c_{0} > 0 $

    for which we do not know whether there is a compact embedding $ \mathcal{S}_{\rho, p}(\Omega) $ into $ L^{p}(\Omega; \mathbb{R}^{d}) $. One such kernel is $ \rho({ \boldsymbol \xi}) = |{ \boldsymbol \xi}|^{p-d} $. Nevertheless, we can prove that the associated space $ \mathcal{S}_{\rho, p}(\Omega) $ is compact in the $ L^{p}_{loc} $ topology, a result which we will state and prove in the appendix.

    For scalar fields, compactness results like those stated above are commonplace for spaces corresponding to special kernels such as the standard fractional Sobolev spaces. In [22, Lemma 2.2], for more general radial and monotone decreasing kernels $ \rho $, condition (1.11) is shown to be sufficient for the compact embedding of the space

    $ \left\{f\in L^{2}(\Omega): \int_{\Omega}\int_{\Omega}\rho( {\bf y}- {\bf x})\frac{| {f( {\bf y}) - f( {\bf x})}|^{2}}{| { {\bf y}- {\bf x}}|^{2}} < \infty\right\}\; $

    in $ L^{2}(\Omega) $. The statement is certainly true for any $ 1\leq p < \infty. $ The proof of [22, Lemma 2.2] actually relies on and modifies the argument used to prove another type of compactness result by Bourgain, Brezis and Mironescu in [6, Theorem 4] that applies criteria involving a sequence of kernels. The argument of [6] uses extensions of functions to $ \mathbb{R}^{d} $ where the monotonicity of $ \rho $ is used in an essential way to control the semi-norm of the extended functions by the original semi-norm. That is, let us introduce a sequence of radial functions $ \rho_{n} $ satisfying

    $ n1,ρn0,Rdρn(ξ)dξ=1,and limn|ξ|>rρn(ξ)dξ=0,r>0. $ (1.12)

    Assuming that for each $ n $, $ \rho_{n} $ is nonincreasing, and if

    $ supn1ΩΩρn(yx)|fn(y)fn(x)|p|yx|p<, $ (1.13)

    then $ \{f_n\} $ is precompact in $ L^{p}(\Omega) $, which is the result of [6, Theorem 4] obtained by showing that (1.13) makes it possible to apply a variant of the Riesz-Fréchet-Kolomogorov theorem [7]. In [22, Lemma 2.2], for a fixed $ \rho $, the condition (1.11) is used to replace the role played by the condition (1.12). In [28, Theorem 1.2], the same result as in [6, Theorem 4] was proved by dropping the monotonicity assumption on $ \rho_{n} $ for $ d\geq 2 $. In addition, the proof in [28] avoids the extension of functions to $ \mathbb{R}^{d} $ but rather shows that the bulk of the mass of each $ {f_{n}} $, that is $ \int_{\Omega} |f_n|^{p} $, comes from the interior and quantifies the contribution near the boundary. As a consequence, if (1.13) holds, then as $ n\to \infty $ there is no mass concentration or leak at the boundary, two main causes of failure of compactness. The compactness results were applied to establish some variational convergence results in [29]. Clearly if one merely replaces scalar functions in (1.13) by vector fields, both compactness results [6, Theorem 4] and [28, Theorem 1.2] will remain true. It turns out the results will remain valid for vector fields even under a weaker assumption. Indeed, following the argument [6, Theorem 4] and under the monotonicity assumption that for $ n $, $ \rho_{n} $ is nonincreasing, it was proved in [20, Theorem 5.1] that if $ {\bf u}_{n} $ is a bounded sequence of vector fields satisfying

    $ supn1ΩΩρn(yx)|D(un)(x,y)|pdydx< $ (1.14)

    then $ \{{\bf u}_{n}\} $ precompact in the $ L^{p}_{loc}(\Omega; \mathbb{R}^{d}) $ topology with any limit point being in $ W^{1, p}(\Omega; \mathbb{R}^{d}) $ when $ 1 < p < \infty $, and in $ BD(\Omega) $ when $ p = 1 $. Here, $ BD(\Omega) $ is the space of functions with bounded deformation. Later, again under the monotonicity assumption on $ \rho_{n} $, but using the argument of [28, Theorem 1.2] instead, it was proved in [25, Proposition 4.2] that in fact, (1.14) implies that $ \{{\bf u}_{n}\} $ is precompact in the $ L^{p}(\Omega; \mathbb{R}^{d}) $ topology. In this paper, we will prove a similar result relaxing the requirement that $ \rho_{n} $ is a Dirac-Delta sequence.

    Theorem 1.4. Let $ \rho \in L^{1}_{loc} $ satisfy (1.10) and (1.11). For each $ n $, $ \rho_{n} $ is radial and $ \rho_{n} $ satisfies (1.10) and that

    $ \rho_{n}\geq 0, \quad \rho_{n}\rightharpoonup \rho, \quad{{weakly\ in\ L^{1}_{loc}(\mathbb{R}^{d}) }}, {{\; \; \; \; and \ \rho_n \leq c \rho }} $

    for some $ c > 0. $ Then, if $ \{{\bf u}_{n}\} $ is a bounded sequence in $ L^{p}(\Omega; \mathbb{R}^{d}) $ such that (1.14) holds, then $ \{{\bf u}_{n}\} $ is precompact in $ L^{p}(\Omega; \mathbb{R}^{d}) $. Moreover, any limit point is in $ \mathcal{S}_{\rho, p}(\Omega). $

    A natural by-product of Theorem 1.4 is the Poincaré-Korn type inequality stated below.

    Corollary 1.5 (Poincaré-Korn type inequality). Suppose that $ 1 \leq p < \infty $ and $ V $ is a weakly closed subset of $ L^{p}(\Omega; \mathbb{R}^{d}) $ such that $ V\cap\mathcal{R} = \{{ \bf 0}\} $. Let $ \rho \in L^{1}_{loc} $ satisfies (1.10) and (1.11). Let $ \rho_{n} $ be a sequence of radial functions, and for each $ n $, $ \rho_{n} $ satisfies (1.10) and that

    $ \rho_{n}\geq 0, \quad \rho_{n}\rightharpoonup \rho, \quad{{weakly\ in\ L^{1}_{loc}(\mathbb{R}^{d}) }},\ {{and \ \rho_n \leq c \rho }} $

    for some $ c > 0 $. Then there exist constants $ C > 0 $ and $ N\geq 1 $ such that

    $ Ω|u|pdxCΩΩρn(yx)|(u(y)u(x))|yx|(yx)|yx||pdydx $ (1.15)

    for all $ {\bf u}\in V\cap L^{p}(\Omega; \mathbb{R}^{d}) $ and $ n\geq N $. The constant $ C $ depends only on $ V, d, p $, $ \rho $, and the Lipschitz character of $ \Omega $.

    We note that the Poincaré-Korn-type inequality has been proved for Dirac-Delta sequence of kernels $ \rho_{n} $ [23,25]. The corollary extends the result to sequence of kernels that weakly converge to a given function $ \rho $ satisfying (1.10) and (1.11).

    The rest of the paper is devoted to prove the main results and it is organized as follows. We prove Theorem 1.2 in Section 2. Theorem 9.1 and Proposition 3.5 are proved in Section 3. The proof of Theorems 1.3 and 1.4 and Corollary 1.5 are presented in Section 4. Further discussions are given at the end of the paper.

    In this section we will prove the $ \Gamma $-convergence of the sequence of energies $ E_{\rho_n} $ defined in (1.5). The proof relies on a sequence of results on the limiting behavior of functions as well as the action of operators. To that end, we assume that $ \rho $ and $ \{\rho_n\} $ satisfy (1.6)–(1.8) throughout this section. We begin with the convergence properties of the nonlocal divergence operator.

    Lemma 2.1. Suppose that $ {\bf u}_n \to {\bf u} $ strongly in $ L^{2}(\Omega; \mathbb{R}^{d}) $, $ {\bf u}\in \mathcal{S}_{\rho, 2}(\Omega) $, and that $ \{\mathfrak{D}_{\rho_n} ({\bf u}_n)\} $ is uniformly bounded in $ L^{2}(\Omega) $. Then $ \mathfrak{D}_{\rho_n}({\bf u}_n) \rightharpoonup \mathfrak{D}_{\rho}({\bf u}) $ weakly in $ L^{2}(\Omega) $.

    Proof. We recall the nonlocal integration by parts formula ([25,26]) that for any $ v\in W^{1, 2}_0(\Omega) $

    $ \int_{\Omega} \mathfrak{D}_{\rho_n}({\bf u}_n) v( {\bf x}) dx = -\int_{\Omega} \mathcal{G}_{n}(v)( {\bf x}) \cdot {\bf u}_{n}( {\bf x}) d {\bf x} $

    where $ \mathcal{G}_{n}(v)({\bf x}) $ is the nonlocal gradient operator

    $ \mathcal{G}_{\rho_n}(v)( {\bf x}) = p.v. \int_{ \Omega} \rho_{n}( {\bf y}- {\bf x})\frac{v( {\bf y})+v( {\bf x})}{| {\bf y}- {\bf x}|} \frac{ {\bf y}- {\bf x}}{| {\bf y}- {\bf x}|}d {\bf y}. $

    Now for $ v\in C^{1}_{c}(\Omega) $, we may rewrite the nonlocal gradient as

    $ \mathcal{G}_{\rho_n}(v)( {\bf x}) = \int_{ \Omega} \rho_{n}( {\bf y}- {\bf x})\frac{v( {\bf y})-v( {\bf x})}{| {\bf y}- {\bf x}|} \frac{ {\bf y}- {\bf x}}{| {\bf y}- {\bf x}|}d {\bf y} + 2 \int_{ \Omega} \rho_{n}( {\bf y}- {\bf x})\frac{v( {\bf x})}{| {\bf y}- {\bf x}|} \frac{ {\bf y}- {\bf x}}{| {\bf y}- {\bf x}|}d {\bf y}. $

    and estimate as [26, Corollary 2.4],

    $ \| \mathcal{G}_{\rho_n}(v)\|_{L^{\infty}} \leq 3 \|\rho_{n}\|_{L^{1}}\|\nabla v\|_{L^{\infty}} \leq C \|\rho\|_{L^{1}}\|\nabla v\|_{L^{\infty}}. $

    Also, it is not difficult to show that for all $ {\bf x}\in \Omega $, $ \mathcal{G}_{\rho_n}(v)({\bf x}) \to \mathcal{G}_{\rho}(v)({\bf x}) $. This follows from the convergence of $ \rho_n $ to $ \rho $ in $ L^{1}_{loc}(\mathbb{R}^d) $. We thus conclude that for $ v\in C^{1}_{c}(\Omega) $,

    $ \mathcal{G}_{\rho_n}(v) \to \mathcal{G}_{\rho}(v)\quad \text{strongly in}\ L^{2} . $

    Thus from the above integration by parts formula we have that for any $ v\in C^{1}_{c}(\Omega) $

    $ limnΩDρn(un)v(x)dx=limnΩGρn(v)(x)un(x)dx=ΩGρ(v)(x)u(x)dx=ΩDρ(u)v(x)dx. $

    The last inequality is possible because $ {\bf u}\in \mathcal{S}_{\rho, 2}(\Omega) $. Now for any $ v\in L^{2}(\Omega) $, let us choose $ v_m\in C^{1}_{c}(\Omega) $ such that $ v_m\to v $ strongly in $ L^{2}(\Omega) $. Then we have for each $ n, m $ that

    $ \int_{\Omega} \mathfrak{D}_{\rho_n}({\bf u}_n) v( {\bf x}) d {\bf x} = \int_{\Omega} \mathfrak{D}_{\rho_n}({\bf u}_n) v_m( {\bf x}) dx + R_{n, m} $

    where

    $ |R_{n, m}| = \left| \int_{\Omega} \mathfrak{D}_{\rho_n}({\bf u}_n) (v( {\bf x})-v_{m}( {\bf x})) d {\bf x}\right|\leq \|\mathfrak{D}_{\rho_n}({\bf u}_n)\|_{L^{2}(\Omega)}\|v_m - v\|_{L^{2}(\Omega)} {.} $

    Therefore using the fact that $ \|\mathfrak{D}_{\rho_n}({\bf u}_n)\|_{L^{2}(\Omega)} $ is uniformly bounded in $ n $, we have that $ \lim_{m\to \infty} \sup_{n\in \mathbb{N}} |R_{n, m}| = 0 $ and so we have

    $ lim infnΩDρn(un)v(x)dx=limnΩDρn(un)vm(x)dx+lim infnRn,m=ΩDρ(u)vm(x)dx+lim infnRn,m. $

    We now take $ m\to \infty $ and use the fact that $ \mathfrak{D}_{\rho}({\bf u})\in L^{2}(\Omega) $ to complete the proof the lemma.

    Lemma 2.2. Suppose that $ {\bf u}_n \to {\bf u} $ strongly in $ L^{2}(\Omega; \mathbb{R}^d) $, $ {\bf u}\in \mathcal{S}_{\rho, 2}(\Omega), $ and that $ \sup_{n\in \mathbb{N}}W_{\rho_n} ({\bf u}_n) \leq C < \infty. $ Then it holds that

    $ ΩΩρ(xx)(D(u)(x,x)1dDρ(u)(x))2dxdxlim infnΩΩρn(xx)(D(un)(x,x)1dDρn(un)(x))2dxdx. $ (2.1)

    Proof. Let $ A\subset \subset \Omega $ and $ \varphi\in C^{\infty}_c(B_1(0)) $. For $ \epsilon < \text{dist}(A, \partial \Omega) $, consider the sequence of functions $ \varphi_\epsilon \ast {\bf u}_n $ and $ \varphi_\epsilon \ast \mathfrak{D}_{\rho_n}({\bf u}_n) $ defined for $ {\bf x}\in A $, where $ \varphi_\epsilon({\bf z}) = \epsilon^{-d}\varphi({\bf z}/\epsilon) $ is standard mollifiers. Then since $ {\bf u}_n \to {\bf u} $ strongly in $ L^{2} $, for a fixed $ \epsilon > 0, $ we have as $ n\to \infty $,

    $ φϵunφϵuin C2(¯A;Rd) andφϵDρn(un)φϵDρ(u)strongly in L2(A). $ (2.2)

    The latter follows from Lemma 2.1 and the fact that the convolution is a compact operator. Using Jensen's inequality, we have that for each $ \epsilon > 0 $ small and $ n $ large

    $ AAρn(xx)(D(φϵun)(x,x)1dφϵDρn(un)(x))2dxdxAAρn(xx)(D(un)(x,x)1dDρn(un)(x))2dxdx. $ (2.3)

    The left hand side of (2.3) can be rewritten after change of variables as

    $ AAρn(xx)(D(φϵun)(x,x)1dφϵDρn(un)(x))2dxdx=Rdρn(z)AχA(x+z)(D(φϵun)(x,x+z)1dφϵDρn(un)(x))2dxdz. $

    Using (2.2), the sequence of functions

    $ {\bf z}\mapsto \int_{A}\chi_{A}( {\bf x}+ {\bf z})\left( {\mathscr D}(\varphi_\epsilon \ast{\bf u}_n)( {\bf x}, {\bf x} + {\bf z}) -\frac{1}{d}\, \varphi_\epsilon \ast\mathfrak{D}_{\rho_n}({\bf u}_n)( {\bf x})\right)^2d {\bf x}\ $

    converges in $ L^{\infty}(\Omega) $ as $ n\to\infty $ to

    $ {\bf z}\mapsto \int_{A}\chi_{A}( {\bf x}+ {\bf z})\left( {\mathscr D}(\varphi_\epsilon \ast{\bf u})( {\bf x}, {\bf x} + {\bf z}) -\frac{1}{d}\, \varphi_\epsilon \ast\mathfrak{D}_{\rho}({\bf u})( {\bf x})\right)^2d {\bf x} $

    where we use the simple inequality $ |a^2 - b^2|\leq ||a| + |b|||a-b| $ and the assumption that $ {\bf u}\in \mathcal{S}_{\rho, 2}(\Omega) $. Using the convergence of $ \rho_n $ to $ \rho $ in $ L^{1}_{loc}(\mathbb{R}^d) $ and taking the limit in (2.3) we conclude that for each $ \epsilon > 0 $,

    $ AAρ(xx)(D(φϵu)(x,x)1dφϵDρ(u)(x))2dxdxlim infnAAρn(xx)(D(un)(x,x)1dDρn(un)(x))2dxdx. $

    Now inequality (2.1) follows after applying first Fatou's lemma in $ \epsilon $ and noting that $ A\subset\subset \Omega $ was arbitrary.

    Let us state some elementary inequalities that relate the energy $ W_{\rho}({\bf u}) $ and its integrand with that of the seminorm $ |{\bf u}|_{\mathcal{S}_{\rho, 2}} $. The proof follows from direct calculations and uses a simple application of Hölder's inequality.

    Lemma 2.3. For a given $ \rho\in L^{1}_{loc}(\mathbb{R}^d) $ and $ \Omega $ bounded such that for $ {\bf u}\in \mathcal{S}_{\rho, 2} (\Omega) $ and $ {\bf x} \in \Omega $ we have

    $ Dρ(u)2(x)ρL1(BR(0))Ωρ(xy)|D(u)(x,y)|2dy,Ωρ(yx)(D(u)(x,y)1dDρ(u)(x))2dyC(d,ρL1(BR(0)))Ωρ(xy)|D(u)(x,y)|2dy. $

    Moreover, we have positive constants $ C_1 $ and $ C_2 $, depending on $ \rho, \Omega, $ and $ d $, such that for all $ {\bf u}\in \mathcal{S}_{\rho, 2} (\Omega) $

    $ C_{1} |{\bf u}|_{\mathcal{S}_{\rho, 2} (\Omega)}^{2} \leq W_{\rho}({\bf u}) \leq C_2 |{\bf u}|_{\mathcal{S}_{\rho, 2} (\Omega)}^{2}. $

    Proof of Theorem 1.2. The proof has two parts: the demonstration of the $ \Gamma $-convergence of the energy functionals and the proof of the convergence of minimizers. For the first part, following the definition of $ \Gamma $-convergence, we prove the two items in Definition 1.1.

    Item a) Suppose that $ {{\bf u}_n} \to {\bf u} $ strongly in $ L^{2} $. We will show that

    $ \overline{E}_{\rho}({\bf u})\leq \liminf\limits_{n\to \infty}\overline{E}_{n}({\bf u}_{n}) {.} $

    Since $ \int_{\Omega}{\bf f}\cdot {\bf u}_nd {\bf x} \to \int_{\Omega}{\bf f}\cdot {\bf u}d {\bf x} $ as $ n\to \infty, $ we only need to show that

    $ {W}_{\rho}({\bf u})\leq \liminf\limits_{n\to \infty}{W}_{\rho_n}({\bf u}_{n}) {.} $

    To that end, we will assume without loss of generality that $ \liminf_{n\to \infty}{W}_{\rho_{n}}({\bf u}_{n}) < \infty $, and so (up to a subsequence) $ \sup_{n\in \mathbb{N}}W_{\rho_n} ({\bf u}_n) \leq C < \infty. $ Then we have that $ \{\mathfrak{D}_{\rho_n}({\bf u}_n)\} $ is uniformly bounded in $ L^{2}(\Omega) $ and $ \{|{\bf u}_{n}|_{\mathcal{S}_{\rho_n, 2}}\} $ is uniformly bounded as well, by Lemma 2.3. To prove the desired inequality it suffices to show that

    $ Ω(Dρ(u))2dxlim infnΩ(Dρn(un))2dx $ (2.4)

    and

    $ ΩΩρ(xx)(D(u)(x,x)1dDρ(u)(x))2dxdxlim infnΩΩρn(xx)(D(un)(x,x)1dDρn(un)(x))2dxdx. $ (2.5)

    To show (2.4), using the weak lower semicontinuity of norm, it suffices to show that $ \mathfrak{D}_{\rho_n}({\bf u}_n) \rightharpoonup \mathfrak{D}_{\rho}({\bf u}) $ weakly in $ L^{2}(\Omega) $. But this is proved in Lemma 2.1 after noting the above assumption.

    Inequality (2.5) will follow from Lemma 2.2 if we show $ {\bf u} \in \mathcal{S}_{\rho, 2}(\Omega) $. But under the assumptions on the sequence $ {\bf u}_n $, the conclusion $ {\bf u} \in \mathcal{S}_{\rho, 2}(\Omega) $ follows from Theorem 1.4 that will be proved in the coming sections.

    Item b) For a given $ {\bf u}\in L^{2}(\Omega) $, we take the recovery sequence to be $ {\bf u}_{n} = {\bf u} $. Now if $ {\bf u}\in L^{2}(\Omega)\setminus \mathcal{S}_{\rho, 2}(\Omega) $, then by definition $ \bar{E}_{\infty}({\bf u}) = \infty $ and necessarily $ \liminf_{n\to \infty} E_{\rho_{n}}({\bf u}) = \infty $. Otherwise, up to a subsequence (nor renamed) $ \sup_{n} E_{\rho_n}({\bf u}) < \infty $ and

    $ [{\bf u}]_{\mathcal{S}_{\rho_n, 2}}^2 \leq C (E_{\rho_{n}}({\bf u}) + \|{\bf u}\|_{L^{2}}) \leq C + \|{\bf u}\|_{L^{2}}, $

    where we used Lemma 2.3. Then by passing to the limit and using Fatou's lemma, we have $ [{\bf u}]_{\mathcal{S}_{\rho, 2}}^2 < \infty $, that is, $ {\bf u}\in \mathcal{S}_{\rho, 2}(\Omega) $, which is a contradiction. In the event that $ {\bf u}\in \mathcal{S}_{\rho, 2}(\Omega) $, we may use (1.8) to get the pointwise convergence and Lemma 2.3 to get appropriate bounds of the integrand of $ W_{\rho_n}({\bf u}) $ to apply the Dominated Convergence Theorem and conclude that $ \liminf_{n\to \infty} W_{\rho_{n}}({\bf u}) = W_{\rho}({\bf u}) $, from which Item b) follows.

    We next prove the second part of the theorem, the convergence of minimizers. To apply [21, Theorem 7.8 and Corollary 7.202], we need to prove the equicoercvity of the functionals restricted to $ V\cap \mathcal{S}_{\rho_n, 2} $. That is, for $ {\bf u}_n \in V\cap \mathcal{S}_{\rho_n, 2} $ such that $ \sup_{n \geq 1} E_{\rho_n}({\bf u}_n) < \infty $, we need to show that the sequence $ \{{\bf u}_{n}\} $ is precompact in $ L^{2}(\Omega; \mathbb{R}^d) $. To that end, first a positive constant $ C > 0 $ and for all $ n\geq 1 $

    $ [{\bf u}_n]_{\mathcal{S}_{\rho_n, 2}}^2 \leq C (E_{\rho_{n}}({\bf u}_n) + \|{\bf u}_n\|_{L^{2}}) \leq C + \|{\bf u}_n\|_{L^{2}}. $

    Using the uniform Poincaré-Korn inequality, Theorem 1.5, for all large $ n $ we have that $ \|{\bf u}_n\|_{L^{2}} \leq C [{\bf u}_n]_{\mathcal{S}_{\rho_n, 2}} $ and as a consequence

    $ [{\bf u}_n]_{\mathcal{S}_{\rho_n, 2}}^2 \leq C(1 + [{\bf u}_n]_{\mathcal{S}_{\rho_n, 2}})\quad \text{for all n large}. $

    It then follows that $ [{\bf u}_n]_{\mathcal{S}_{\rho_n, 2}} $ is uniformly bounded and therefore, by the uniform Poincaré-Korn inequality, $ \|{\bf u}_n\|_{L^{2}(\Omega)} $ is bounded as well. We now use the compactness result, Theorem 1.4, to conclude that $ \{{\bf u}_n\} $ is precompact in $ L^{2}(\Omega; \mathbb{R}^2) $ with limit point $ {\bf u} $ in $ \mathcal{S}_{\rho, 2}(\Omega) \cap V. $ We may now apply [21, Theorem 7.8 and Corollary 7.202] to state that $ {\bf u} $ is a minimizer of $ E_{\rho} $ over $ \mathcal{S}_{\rho, 2}(\Omega) \cap V. $

    The proof of the $ L^{p} $ compactness result, Theorem 1.3, will be carried out in two steps. We establish first compactness in $ L^{p}_{loc} $ topology followed by proving a boundary estimate that controls growth near the boundary of the domain. The $ L^{p}_{loc} $ compactness will be proved in this section under a weaker assumption on the kernel. In fact $ L^{p}_{loc} $ compactness will be stated and proved for a broader class of kernels that include kernels of the type $ \tilde{\rho} ({ \boldsymbol\xi})\chi_{B_{1}^{\Lambda}}({ \boldsymbol\xi}) $ where $ \tilde{\rho} $ satisfies (1.10) and (1.11), where $ B_{1}^{\Lambda} = \{ {\bf x}\in B_{1}: { {\bf x}/|{ {\bf x}}|}\in \Lambda\} $ is a conic region spanned by a given a nontrivial spherical cap $ \Lambda \subset\mathbb{S}^{d-1} $. To make this and the condition of the theorem precise, we begin identifying the kernel $ \rho $ by the representative

    $ {\rho}( {\bf x}) = \left\{limh0Bh(x)ρ(ξ)dξ,if x is a Lebesgue point,,otherwise.\right. $

    For $ \theta_{0}\in (0, 1) $ and $ {\bf v}\in \mathbb{S}^{d-1} $, let us define

    $ \rho_{\theta_{0}}(r{\bf v}) = \inf\limits_{\theta\in [\theta_{0}, 1]}\rho(\theta r{\bf v})\theta^{-p}. $

    It is clear that for a given $ {\bf v}\in \mathbb{S}^{d-1} $, $ \rho_{\theta_{0}}(r{\bf v}) \leq \rho(\theta r{\bf v})\theta^{-p} $ for any $ \theta\in [\theta_{0}, 1] $ and $ r\in (0, \infty) $. In particular, this implies $ \rho_{\theta_{0}}({ \boldsymbol \xi}) \leq \rho({ \boldsymbol \xi}) $ for any $ { \boldsymbol \xi} $, with the equality holds if $ \rho $ is radial and $ |{ \boldsymbol \xi}|^{-p}\rho({ \boldsymbol \xi}) $ is nonincreasing in $ |{ \boldsymbol \xi}| $.

    We now make a main assumption on $ \rho $ that

    $ θ0(0,1),ΛSd1 and v0Λ such that Hd1(Λ)>0,ρθ0(rv)=ρθ0(rv0),(r,v)(0,)×Λ,andlimδ0δpδ0ρθ0(rv0)rd1dr=0. $ (3.1)

    Assumption (3.1) says that, on a conic region with apex at the origin, the kernel $ \rho $ is above a nonnegative function with appropriate singular growth near the origin. Note that on one hand, it is not difficult to see if $ \rho \in L^{1}_{loc}(\mathbb{R}^{d}) $ is a nonnegative function that satisfies (1.10) and (1.11), then it also satisfies (3.1). On the other hand, if $ \tilde{\rho} $ satisfies (1.10) and (1.11), then given a nontrivial spherical cap $ \Lambda $ and conic region $ B_{1}^{\Lambda} = \{ {\bf x}\in B_{1}: { {\bf x}/|{ {\bf x}}|}\in \Lambda\} $, the kernel $ \rho({ \boldsymbol \xi}) = \tilde{\rho} ({ \boldsymbol \xi})\chi_{B_{1}^{\Lambda}}({ \boldsymbol \xi}) $ satisfies (3.1) (with $ \theta_0 $ being any number in $ (0, 1) $ and $ {\bf v}_0 $ representing any vector in $ \Lambda $) but not necessarily (1.10) and (1.11). For kernels of this form, we need the formulation in (3.1) to carry out the proof of $ L^{p}_{loc} $ compactness. We should also note that one can construct other $ \rho $ that are not of the above form that satisfy (3.1), see [6, Eq (17)].

    Theorem 3.1 ($ L^{p}_{loc} $ compactness). Suppose that $ 1\leq p < \infty $. Let $ \rho \in L^{1}(\mathbb{R}^{d}) $ be a nonnegative function satisfying (3.1). Suppose also that $ \{{\bf u}_{n}\} $ is a sequence of vector fields that is bounded in $ \mathcal{S}_{\rho, p}(\mathbb{R}^{d}) $. Then for any $ D\subset \mathbb{R}^{d} $ open and bounded, the sequence $ \{{\bf u}_{n}|_{D}\} $ is precompact in $ L^{p}(D; \mathbb{R}^{d}) $.

    We should mention that although the focus is different, operators that use non-symmetric kernels like those satisfying the condition (3.1) have been studied in connection with semi-Dirichlet forms and the processes they generate, see [2,16] for more discussions. In particular, most of the examples of kernels listed in [16, Section 6] satisfy condition (3.1).

    We begin with the following lemma whose proof can be carried out following the argument used in [28]. Let $ {\bf u}\in L^{p}(\mathbb{R}^{d}; \mathbb{R}^{d}) $ be given, we introduce the function $ F_{p}[{\bf u}]:\mathbb{R}^{d}\to [0, \infty) $ defined by

    $ F_{p}[{\bf u}]({\bf h}) = \int_{\mathbb{R}^{d}}\left|({\bf u}( {\bf x} + {\bf h}) - {\bf u}( {\bf x}))\cdot \frac{{\bf h}}{|{\bf h}|}\right|^{p}d{ {\bf x}}, \quad \text{for}\ {\bf h}\in \mathbb{R}^{d} . $

    Lemma 3.2. Suppose that $ \theta_{0} $ is given as in (3.1). There exists a constant $ C = C(\theta_{0}, p) > 0 $ such that for any $ \delta > 0 $, and $ {\bf v}\in \mathbb{S}^{d-1} $

    $ Fp[u](tv)Cδpδ0ρθ0(sv)sd1ds0ρ(hv)hd1Fp[u](hv)hpdh, $

    for any $ 0 < t < \delta $ and any $ {\bf u}\in L^{p}(\mathbb{R}^{d}, \mathbb{R}^{d}) $.

    Proof. For any $ {\bf v}\in \mathbb{S}^{d-1} $ and $ t\in \mathbb{R} $, we may rewrite the function $ F_{p} $ as

    $ F_{p}[{\bf u}](t{\bf v}) = \int_{\mathbb{R}^{d}}|({\bf u}( {\bf x} + t{\bf v}) - {\bf u}( {\bf x}))\cdot {\bf v}|^{p}d{ {\bf x}}. $

    It follows from [28, Lemma 3.1] that given $ 0 < s < t $, there exist $ C_{p} $ and $ \theta = \frac{t}{s} - k\in (0, 1) $ ($ k $ an integer) such that

    $ \frac{F_{p}[{\bf u}](t{\bf v})}{ t^{p}} \leq C_{p}\left\{ {F_{p}[{\bf u}](s{\bf v})\over s^{p}} + {F_{p}[{\bf u}](\theta s{\bf v})\over t^{p}}\right\}. $

    We also have that for a given $ l_{0} \in \mathbb{N} $,

    $ F_{p}[{\bf u}](\theta s{\bf v}) \leq l_{0}^{p} F_{p}[{\bf u}]\left(\frac{\theta s}{ l_{0}} {\bf v}\right) \leq {2^{(p-1)}}l^{p}_{0}\left\{ F_{p}[{\bf u}](s{\bf v}) + F_{p}[{\bf u}]\left(s - \frac{s\theta}{{l_{0}}}{\bf v}\right)\right\}. $

    Combining the above we have that for any $ l_{0} $, there exists a constant $ C = C(p, l_{0}) $ such that

    $ Fp[u](tv)tpC(p,l0){Fp[u](sv)sp+Fp[u](˜θsv)tp},where ˜θ=1θl0. $ (3.2)

    Now let us take $ \theta_{0} $ as given in (3.1) and choose $ l_{0} $ large that $ {1\over l_{0}} < 1-\theta_{0} $. It follows that $ \theta_{0} < \tilde{\theta} \leq 1 $. Then for any $ \delta > 0 $, and any $ 0 < s < \delta \leq \tau $, by multiplying both sides of inequality (3.2) by $ \rho_{\theta_{0}} ({\bf v}s) $ and integrating from $ 0 $ to $ \delta $, we obtain

    $ δ0ρθ0(sv)sd1dsFp[u](τv)τpC(p,l0){δ0ρθ0(sv)sd1Fp[u](sv)spds+δ0ρθ0(sv)sd1Fp[u](˜θsv)τpds}. $

    Let us estimate the second integral in the above:

    $ I = {1\over \tau^{p}}\int_{0}^{\delta} \rho_{\theta_{0}}(s{\bf v}) s^{d-1}{F_{p}[{\bf u}](\tilde{\theta} s{\bf v})}ds. $

    We first note that using the definition of $ \rho_{\theta_{0}} $ and since $ \delta \leq \tau $, we have

    $ I\leq {1 \over {\theta_{0}^{d-1}}} \int_{0}^{\tau} \rho(\tilde{\theta}s{\bf v}) (\tilde{\theta} s)^{d-1}{F_{p}[{\bf u}](\tilde{\theta} s{\bf v})\over {(\tilde{\theta} s)^{p}}}ds. $

    The intention is to change variables $ h = \tilde{\theta} s $. However, note that $ \tilde{\theta} $ is a function of $ s $, and by definition

    $ \tilde{\theta}s = \left(\frac{k}{ l_{0}} + 1\right)s - \frac{\tau}{ l_{0}}\quad \text{for}\ k \leq \frac{\tau}{s} < k + 1 . $

    It then follows by a change of variables that

    $ I1θd10k=1τkτ(k+1)ρ(˜θsv)(˜θs)d1Fp[u](˜θsv)(˜θs)pds=1θd10k=1τkτ(11l0)(k+1)ρ(hv)hd1Fp[u](hv)hpdhkl0+1C0ρ(hv)hd1Fp[u](hv)hpdh, $

    where in the last estimate integrals over overlapping domains were counted at most a finite number of times. Combining the above estimates we have shown that there exists a constant $ C $ such that for any $ {\bf v} \in \mathbb{S}^{d-1} $, $ \delta > 0 $ and $ \tau\geq \delta $

    $ \left(\int_{0}^{\delta} \rho_{\theta_{0}} (s{\bf v})s^{d-1} ds\right) \frac{F_{p}[{\bf u}](\tau{\bf v})}{ \tau^{p}} \leq C \int_{0}^{\infty}\rho(h{\bf v}) h^{d-1}\frac{F_{p}[{\bf u}](h{\bf v})}{ {h^{p}}}dh. $

    Rewriting the above and restricting $ {\bf v} \in \Lambda $ we have that

    $ F_{p}[{\bf u}](\tau{\bf v}) \leq C \frac{\tau^{p}}{ \int_{0}^{\delta} \rho_{\theta_{0}} (s{\bf v})s^{d-1} ds} \int_{0}^{\infty}\rho(h{\bf v}) h^{d-1}\frac{F_{p}[{\bf u}](h{\bf v})}{h^{p}}dh. $

    Now let $ 0 < t < \delta $ and applying the above inequality for $ \tau = \delta $ and $ \tau = t + \delta $, we obtain

    $ Fp[u](tv)=Fp[u]((t+δ)vδv)2p1{Fp[u]((t+δ)v)+Fp[u](δv)}Cδpδ0ρθ0(sv)sd1ds0ρ(hv)hd1Fp[u](hv)hpdh. $

    This completes the proof.

    Lemma 3.3. Suppose that $ \rho\in L^{1}_{loc}(\mathbb{R}^{d}) $ and there exists a spherical cap $ \Lambda\subset \mathbb{S}^{d-1} $ and a vector $ {\bf v}_{0}\in\Lambda $ such that the function $ \rho(r{\bf v}) = \rho(r{\bf v}_{0}) = \tilde{\rho}(r) $, for all $ {\bf v}\in \Lambda $ and $ r\mapsto r^{-p}\tilde{\rho}(r) $ is nonincreasing. Then there exists a constant $ C = C(d, p, \Lambda) $ such that for any $ \delta > 0 $, and $ {\bf v}\in \Lambda $,

    $ Fp[u](tv)Cδpδ0˜ρ(s)sd1ds0ρ(hv)hd1Fp[u](hv)hpdh, $

    for any $ 0 < t < \delta $ and any $ {\bf u}\in L^{p}(\mathbb{R}^{d}, \mathbb{R}^{d}) $.

    Proof. It suffices to note that for $ \rho\in L^{1}_{loc}(\mathbb{R}^{d}) $ that satisfies the conditions in the statement of the proposition, we have that for any $ \theta_{0} \in (0, 1) $, and any $ {\bf v}\in \Lambda $,

    $ \rho_{\theta_{0}} (r{\bf v}) = r^{p} \inf\limits_{\theta\in [\theta_{0}, 1]} \rho(\theta r{\bf v})(\theta r)^{-p} = \rho(r{\bf v}) = \rho(r{\bf v}_{0}) = \tilde{\rho}(r). $

    We may then repeat the argument in the proof of Lemma 3.2.

    Before proving one of the main results, we make an elementary observation.

    Lemma 3.4. Let $ 1\leq p < \infty. $ Given a spherical cap $ \Lambda $ with aperture $ \theta $, there exists a positive constant $ c_{0}, $ depending only on $ d, \theta $ and $ p $, such that

    $ infwSd1ΛSd1|ws|pdσ(s)c0>0. $

    The above lemma follows from the fact that the map

    $ {\bf w}\mapsto \int_{\Lambda\cap\mathbb{S}^{d-1}} |{\bf w}\cdot {\bf s}|^{p}d\sigma({\bf s}) $

    is continuous on the compact set $ \mathbb{S}^{d-1}, $ and is positive, for otherwise the portion of the unit sphere $ \Lambda $ will be orthogonal to a fixed vector which is not possible since $ \mathcal{H}^{d-1}(\Lambda) > 0 $.

    From the assumption we have

    $ supn1unpLp+supn1RdRdρ(xx)|D(un)(x,x)|pdxdx<. $ (3.3)

    We will use the compactness criterion in [20, Lemma 5.4], which is a variant of the well-known Riesz-Fréchet-Kolmogorov compactness criterion [7, Chapter Ⅳ.27]. Let $ \Lambda $ be as given in (3.1). For $ \delta > 0 $, let us introduce the matrix $ Q = (q_{ij}) $, where

    $ q_{ij} = \int_{\Lambda} s_{i}s_{j} d \mathcal{H}^{d-1}({\bf s}). $

    The symmetric matrix $ \mathbb{Q} $ is invertible. Indeed, the smallest eigenvalue is given by

    $ \lambda_{min} = \min\limits_{| {\bf x}| = 1} \langle \mathbb{Q} {\bf x}, {\bf x} \rangle = \min\limits_{| {\bf x}| = 1} \int_{\Lambda} | {\bf x}\cdot {\bf s}|^{2} d \mathcal{H}^{d-1}({\bf s}) $

    which we know is positive by Lemma 3.4. We define the following matrix functions

    $ P(z)=dQ1zz|z|2χBΛ1(z),Pδ(z)=δdP(zδ) $

    where $ B_{1}^{\Lambda} = \{ {\bf x}\in B_{1}: { {\bf x}/|{ {\bf x}}|}\in \Lambda\} $, as defined before. Then for any $ \delta > 0 $,

    $ \int_{\mathbb{R}^{d}} \mathbb{P}^{\delta}({\bf z}) d{\bf z} = \mathbb{I}. $

    To prove the theorem, using [20, Lemma 5.4], it suffices to prove that

    $ limδ0lim supnunPδunLp(Rd)=0. $ (3.4)

    We show next that the inequality (3.3) and condition (3.1) imply (3.4). To see this, we begin by introducing the notation $ B_{\delta}^{\Lambda} = \{ {\bf x}\in B_{\delta}({ \bf 0}): { {\bf x}/|{ {\bf x}}|}\in \Lambda\} $ and applying Jensen's inequality to get

    $ Rd|un(x)Pδun(x)|pdxRd|RdPδ(yx)(un(y)un(x))dy|pdxRd||Λ|Q1BΛδ(x)(yx)|yx|(un(y)un(x))(yx)|yx|dy|pdx|Λ|pQ1pRd|BΛδ(x)(yx)|yx|(un(y)un(x))(yx)|yx|dy|pdx|Λ|pQ1p|BΛδ|δ0Λτd1Fp[un](τv)dHd1(v)dτC(d,p)|BΛδ|δ0Λτd1Fp[un](τv)dHd1(v)dτ $ (3.5)

    where as defined previously

    $ {F}_{p}[{\bf u}_{n}](\tau {\bf v}) = \int_{\mathbb{R}^{d}} \left|{\bf v}\cdot({\bf u}_{n}( {\bf x}+\tau {\bf {\bf v}}) - {\bf u}_{n}( {\bf x}))\right|^{p}d {\bf x}. $

    Moreover, the fact that $ |\Lambda|^{p}\|\mathbb{Q}^{-1}\|^{p} \leq C(d, p, \Lambda) $ for any $ \delta > 0 $ is also used. We can now apply Lemma 3.2 and use the condition (3.1) to obtain that

    $ C(d,p,λ)|BΛδ|δ0Λτd1Fp[un](τv)dHd1(v)dτC(d,p,Λ)|BΛδ|δ0τd1dτΛ(δpδ0ρθ0(sv0)sd1ds0ρ(hv)hd1Fp[un](hv)hpdh)dHd1(v)C(d,p,Λ)δpδ0ρθ0(sv0)sd1ds|un|Sρ,p(Rd). $

    Therefore from the boundedness assumption (3.3) we have,

    $ Rd|un(x)Pδun(x)|pdxC(p,d,Λ)δpδ0ρθ0(sv0)sd1ds. $

    Equation (3.4) now follows from condition (3.1) after letting $ \delta\to 0 $. That completes the proof.

    A corollary of the compactness result, Theorem 3.1, is the following result that uses a criterion involving a sequence of kernels. The effort made in the proof above was to show the theorem for kernel $ \rho $ satisfying (3.1), but the proposition below limits to those satisfying (1.10) and (1.11).

    Proposition 3.5. Let $ \rho \in L^{1}_{loc} $ satisfy (1.10) and (1.11). Let $ \rho_{n} $ be a sequence of radial functions satisfying (1.10) and that $ \rho_{n}\rightharpoonup \rho $ weakly in $ L^{1} $ as $ n\to \infty $. If

    $ \sup\limits_{n\geq 1}\{ \|{\bf u_{n}}\|_{L^{p}(\mathbb{R}^{d})} + |{\bf u}_{n}|_{\mathcal{S}_{\rho_{n}, p}}\} < \infty $

    then $ \{{\bf u}_{n}\} $ is precompact in $ L^{p}_{loc}(\mathbb{R}^{d}; \mathbb{R}^{d}) $. Moreover, if $ A\subset \mathbb{R}^{d} $ is a compact subset, the limit point of the sequence restricting to $ A $ is in $ \mathcal{S}_{\rho, p}(A). $

    Proof. Using Lemma 3.3 applied to each $ \rho_{n} $, we can repeat the argument in the proof of Theorem 3.1 to obtain

    $ \int_{\mathbb{R}^{d}}|{\bf u}_{n}( {\bf x}) - \mathbb{P}^{\delta}*{\bf u}_{n}( {\bf x})|^{p}d {\bf x} \leq C(p, d) \frac{\delta^{p}}{ \int_{0}^{\delta} \rho_{n}(r)r^{d-1} dr } \leq C(p, d) \frac{\delta^{p}}{ \int_{B_{\delta}} \rho_{n}({ \boldsymbol \xi}) d{ \boldsymbol \xi} } \, . $

    Now since $ \rho_{n}\rightharpoonup \rho $, weakly in $ L^{1} $ as $ n\to \infty $, for a fixed $ \delta > 0 $, it follows that

    $ \limsup\limits_{n\to \infty} \int_{\mathbb{R}^{d}}|{\bf u}_{n}( {\bf x}) - \mathbb{P}^{\delta}*{\bf u}_{n}( {\bf x})|^{p}d {\bf x} \leq C(p, d) \frac{\delta^{p}}{ \int_{B_{\delta}} \rho({ \boldsymbol \xi}) d{ \boldsymbol \xi} } \, . $

    We now let $ \delta \to 0 $, and use the assumption (1.11) to obtain

    $ \lim\limits_{\delta\to 0}\limsup\limits_{n\to \infty} \int_{\mathbb{R}^{d}}|{\bf u}_{n}( {\bf x}) - \mathbb{P}^{\delta}*{\bf u}_{n}( {\bf x})|^{p}d {\bf x} = 0, $

    from which the compactness in the $ L^{p}_{loc} $ topology follows.

    We next prove the final conclusion of the proposition. To that end, let $ A \subset \mathbb{R}^{d} $ be a compact subset. For $ \phi\in C_{c}^{\infty}(B_{1}) $, we consider the convoluted sequence of function $ \phi_{\epsilon} *{\bf u}_{n} $, where $ \phi_{\epsilon}({\bf z}) = \epsilon^{-d} \phi({\bf z}/\epsilon) $ is the standard mollifier. Since $ {\bf u}_{n} \to {\bf u} $ strongly in $ L^{p}(A; \mathbb{R}^{d}) $ for a fixed $ \epsilon > 0 $, we have as $ n\to\infty $,

    $ ϕϵunϕϵuin C2(A;Rd). $ (3.6)

    Using Jensen's inequality, we obtain that for any $ \epsilon > 0 $, and $ n $ large,

    $ AAρn(yx)|(ϕϵun(y)ϕϵun(x))(yx)|yx|2|pdydxRdRdρn(yx)|(un(y)un(x))(yx)|yx|2|pdydx. $

    Taking the limit in $ n $ for fixed $ \epsilon $, we obtain for any $ A $ compact that

    $ \int_{A}\int_{A} \rho( {\bf y}- {\bf x}) \left| \frac{(\phi_{\epsilon} *{\bf u} ( {\bf y}) - \phi_{\epsilon} *{\bf u} ( {\bf x})) \cdot ( {\bf y} - {\bf x})}{| {\bf y} - {\bf x}|^{2}}\right|^{p} d {\bf y} d {\bf x} \leq \sup\limits_{n\geq 1} {|{\bf u}_{n}|_{\mathcal{S}_{\rho_{n}, p}}^{p}} < \infty $

    where we have used (3.6) and the fact that $ \rho_{n} $ converges weakly to $ \rho $ in $ L^{1} $. Finally, let $ \epsilon \to 0 $ and use Fatou's lemma (since $ \phi_{\epsilon} *{\bf u}\to {\bf u} $ almost everywhere) to obtain that for any compact set $ A $,

    $ \int_{A}\int_{A} \rho( {\bf y}- {\bf x}) \left| \frac{({\bf u} ( {\bf y}) - {\bf u} ( {\bf x})) \cdot ( {\bf y} - {\bf x})} {| {\bf y} - {\bf x}|^{2}}\right|^{p} d {\bf y} d {\bf x} \leq \sup\limits_{n\geq 1} {|{\bf u}_{n}|_{\mathcal{S}_{\rho_{n}, p}}^{p}} < \infty, $

    hence completing the proof.

    In this section we prove Theorem 1.3. We follow the approach presented in [28]. The argument relies on controlling the $ L^{p} $ mass of each $ {{\bf u}_{n}} $, $ \int_{\Omega} |{\bf u}_{n}| ^{p} d {\bf x} $, near the boundary by using the bound on the seminorm to demonstrate that in the limit when $ n\to \infty $ there is no mass concentration or loss of mass at the boundary. This type of control has been done for the sequence of kernels that converge to the Dirac Delta measure in the sense of measures. We will do the same for a fixed locally integrable kernel $ \rho $ satisfying the condition (1.11).

    In order to control the behavior of functions near the boundary by the semi-norm $ |\cdot|_{\mathcal{S}_{p, \rho}} $, we first present a few technical lemmas.

    Lemma 4.1. [28] Suppose that $ 1\leq p < \infty $ and that $ g\in L^{p}(0, \infty) $. Then there exists a constant $ C = C(p) $ such that for any $ \delta > 0 $ and $ t\in (0, \delta) $

    $ \int_{0}^{\delta} |g(x)|^{p} dx \leq C \delta^{p} \int_{0}^{2\delta} \frac{|g(x +t )-g(x)|^{p}}{t^{p}} dx + 2^{p-1}\int_{\delta}^{3\delta} |g(x)|^{p} dx. $

    Proof. For a given $ t\in (0, \delta) $, choose $ k $ to be the first positive integer such that $ kt > \delta $. Observe that $ (k-1)t \leq \delta $, and so $ kt \leq 2\delta $. Now let us write

    $ |g(x)|p2p1(|g(x+kt)g(x)|p+|g(x+kt)|p)2p1kp1k1j=0|g(x+jt+t)g(x+jt)|p+2p1|g(x+kt)|p. $

    We now integrate in $ x $ on both side over $ (0, \delta) $ to obtain that

    $ δ0|g(x)|pdx2p1kp1k1j=0δ0|g(x+jt+t)g(x+jt)|pdx+2p1δ0|g(x+kt)|pdx2p1kp1k1j=0δ+jtjt|g(x+t)g(x)|pdx+2p13δδ|g(x)|pdx2p1kp2δ0|g(x+t)g(x)|pdx+2p13δδ|g(x)|pdx. $

    Recalling that $ kt \leq 2\delta $, we have that $ k^{p} \leq 2^{p}\delta^{p}/t^{p} $ and we finally obtain the conclusion of the lemma with $ C = 2^{2p-1} $.

    The above lemma will be used on functions of type $ t\mapsto {\bf u}({\bf x} + t{\bf v})\cdot {\bf v} $, for $ {\bf v}\in \mathbb{S}^{d-1} $. Before doing so, we need to make some preparation first. Observe that since $ \Omega $ is a bounded open subset of $ \mathbb{R}^{d} $ with a Lipschitz boundary, there exist positive constants $ r_{0} $ and $ \kappa $ with the property that for each point $ { \boldsymbol \xi}\in \partial \Omega $ there corresponds a coordinate system $ ({\bf x}', x_{d}) $ with $ {\bf x}'\in \mathbb{R}^{d-1} $ and $ x_{d}\in\mathbb{R} $ and a Lipschitz continuous function $ \zeta:\mathbb{R}^{d-1}\to \mathbb{R} $ such that $ |\zeta({\bf x}')-\zeta({\bf y}')|\leq \kappa| {\bf x}'- {\bf y}'| $,

    $ ΩB(ξ,4r0)={(x,xd):xd>ζ(x)}B(ξ,4r0), $

    and $ \partial\Omega\cap B({ \boldsymbol \xi}, 4r_{0}) = \{({\bf x}', x_{d}): x_{d} = \zeta({\bf x}')\}\cap B({ \boldsymbol \xi}, 4r_{0}). $ It is well known that a Lipschitz domain has a uniform interior cone $ \Sigma({ \boldsymbol \xi}, \theta) $ at every boundary point $ { \boldsymbol \xi} $ such that $ B({ \boldsymbol \xi}, 4\, r_{0})\cap \Sigma({ \boldsymbol \xi}, \theta)\subset \Omega. $ The uniform aperture $ \theta \in(0, \pi) $ of such cones depends on the Lipschitz constant $ \kappa $ of the local defining function $ \zeta, $ and does not depend on $ { \boldsymbol \xi}. $ It is not difficult either to see that for any $ r\in (0, 4r_{0}), $ if $ {\bf y} \in B_{r}({ \boldsymbol \xi}), $ then

    $ dist(y,Ω)=inf{|y(x,xd)|:(x,xd)B3r(ξ),xd=ζ(x)}. $

    We now begin to work on local boundary estimates. To do that without loss of generality, see Figure 1 below, after translation and rotation (if necessary) we may assume that $ { \boldsymbol \xi} = { \bf 0} $ and

    $ ΩB(0,4r0)={(x,xd):xd>ζ(x)}B(0,4r0), $
    Figure 1.  $ \Sigma $ is the cone with aperture $ \pi/4 $ (depicted by the blue lines). The Lipschitz graph $ \zeta $ remains outside the double cone with aperture $ \pi/2 - \arctan(1/2) $ (depicted by the red lines). The red dashed line has length $ r $.

    where $ \zeta({ \bf 0}')=0, $ and $ |\zeta({\bf x}')-\zeta({\bf y}')|\leq \kappa| {\bf x}'- {\bf y}'| $. We also assume that the Lipschitz constant $ \kappa = 1/2 $ and the uniform aperture $ \theta = {\pi}/{4}. $ As a consequence, $ \zeta({\bf x}')\leq | {\bf x}'|/2 $ for all $ {\bf x}'\in B_{4r_{0}}({ \bf 0}'). $ Given any $ 0 < r < r_{0}, $ we consider the graph of $ \zeta $:

    $ Γr:={x=(x,ζ(x))Rd:xBr(0)}. $

    We denote the upper cone with aperture $ {\pi}/{4} $ by $ \Sigma $ and is given by

    $ Σ={x=(x,xd)Rd:|x|xd}. $

    Finally we define $ \Omega_{\tau} = \{ {\bf x}\in \Omega: \text{dist}({\bf x}, \partial\Omega) > \tau\} $ to be the set of points in $ \Omega $ at least $ r $ units away from the boundary. Based on the above discussion we have that for any $ r\in (0, r_{0}], $

    $ ΩBr/2Γr+(ΣBr)ΩB3r. $ (4.1)

    Indeed, let us pick $ {\bf x} = ({\bf x}', x_{d})\in \Omega\cap B_{r/2} $. The point $ \boldsymbol{{ \boldsymbol \xi}} = {\bf x}- ({\bf x}', \zeta({\bf x}')) = ({ \bf 0}', x_{d}-\zeta({\bf x}'))\in \Sigma, $ since $ {0 < } x_{d}-\zeta({\bf x}'). $ Moreover, by the bound on the Lipschitz constant $ |\boldsymbol{{ \boldsymbol \xi}}| = |x_{d}-\zeta({\bf x}')| < r/2 + r/4 < r. $ On the other hand, for any

    $ {\bf x} = ( {\bf x}_{1}', \zeta( {\bf x}'_{1})) + ( {\bf x}'_{2}, (x_{2})_{d})\in \Gamma_{r} + ( {\Sigma} \cap B_{r}), $

    we have

    $ \zeta( {\bf x}_{1}' + {\bf x}'_{2})-\zeta( {\bf x}_{1}') \leq {| {\bf x}'_{2}|}/{2} \leq {(x_{2})_{d}}/{2}, $

    showing that $ \zeta({\bf x}_{1}' + {\bf x}'_{2}) < \zeta({\bf x}_{1}') + (x_{2})_{d} $ and therefore $ {\bf x}\in \Omega $. It easily follows that $ {\bf x}\in B_{3r}, $ as well.

    For any $ r\in (0, r_0) $, and $ {\bf x}'\in B_{r} ({ \bf 0}') $, and $ {\bf v} \in \Sigma\cap\mathbb{S}^{d-1} $, $ \text{dist}(({\bf x}', \zeta({\bf x}')) +r {\bf v}, \partial\Omega)\geq r/\sqrt{10} $. Indeed, $ \text{dist}(({\bf x}', \zeta({\bf x}')) +r {\bf v}, \partial\Omega) $ is larger than or equal to the length of the black dashed line, which is larger than or equal to $ r\sin(\pi/4-\arctan(1/2)) = r/\sqrt{10} $.

    In this subsection we establish the near boundary estimate in the following lemma.

    Lemma 4.2. Suppose that $ \Omega\subset \mathbb{R}^{d} $ is a domain with Lipschitz boundary. Let $ 1\leq p < \infty $. Then there exist positive constants $ C_{1}, C_{2} $, $ r_{0} $ and $ \epsilon_{0}\in (0, 1) $ with the property that for any $ r\in (0, r_{0}), $ $ {\bf u}\in L^{p}(\Omega; \mathbb{R}^{d}), $ and any nonnegative and nonzero $ \rho\in L^{1}_{loc}(\mathbb{R}^{d}) $ that is radial, we have

    $ Ω|u|pdxC1(r)Ωϵ0r|u|pdx+C2rpBr(0)ρ(h)dhΩΩρ(xy)|D(u)(x,y)|pdxdy. $

    The constant $ C_{1} $ may depend on $ r $ but the other constants $ C_{2} $ and $ r_{0} $ depend only on $ d, p $ and the Lipschitz constant of $ \Omega $. Here for any $ \tau > 0 $, we define $ \Omega_{\tau} = \{ {\bf x}\in \Omega: \mathit{\text{dist}}({\bf x}, \partial\Omega) > \tau\} $.

    Proof. Following the above discussion, let us pick $ \overline{ \boldsymbol \eta}\in \partial \Omega $ and assume without loss of generality that $ \overline{ \boldsymbol \eta} = { \bf 0}, $ the function $ \zeta $ that defines the boundary $ \partial \Omega $ has a Lipschitz constant not bigger than $ 1/2 $ and the aperture is $ \pi/4. $ Assume first that $ {\bf u} \in L^{p}(\Omega; \mathbb{R}^{d}) $, and vanishes on $ \Omega_{r/\sqrt{10}} $. Let us pick $ \boldsymbol{{ \boldsymbol \xi}} = ({\bf x}', \zeta({\bf x}')) $ such that $ | {\bf x}'| < r $ and $ {\bf v}\in \Sigma \cap \mathbb{S}^{d-1}. $ Let us introduce the function

    $ gξv(t)=u(ξ+tv)v,t(0,3r0). $

    Then for all $ { \boldsymbol \xi}\in \Gamma_{r} $ and $ {\bf v}\in \Sigma\cap\mathbb{S}^{d-1} $, $ { \boldsymbol \xi} + r{\bf v}\in \Omega_{r/\sqrt{10}} $. It follows that, by assumption on the vector field $ {\bf u}, $ the function $ g_{\bf v}^{{ \boldsymbol { \boldsymbol \xi}}}(t) \in L^{p}(0, 2r) $ and $ g_{\bf v}^{ \boldsymbol { \boldsymbol \xi}}(t) = 0 $ for $ t\in (r, 2r). $ We then apply Lemma 4.1 to get a constant $ C_{p} > 0 $ such that for any $ t\in(0, r) $,

    $ r0|u(ξ+sv)v|pdsCrpr0|(u(ξ+sv+tv)u(ξ+sv))v|ptpds, $

    where we used the fact that $ u $ vanishes on $ \Omega_{r/\sqrt{10}} $. Noting that $ \boldsymbol{{ \boldsymbol \xi}} = ({\bf x}', \zeta({\bf x}')) $ for some $ {\bf x}'\in B'_{r}\subset \mathbb{R}^{d-1}, $ we integrate first in the above estimate with respect to $ {\bf x}'\in B'_{r} $ to obtain that

    $ Brr0|u(ξ+sv)v|pdsdxCrpBrr0|(u(ξ+sv+tv)u(ξ+sv))v|ptpdsdx. $

    The next step involves a change of variable $ {\bf y} = ({\bf x}', \zeta({\bf x}')) + s{\bf v} $. Define a mapping $ G: ({\bf x}', s) \mapsto ({\bf x}', \zeta({\bf x}')) + s{\bf v} $. Then the Jacobian of the mapping is defined almost everywhere and is given by

    $ J_G = {\bf v} \cdot (-\nabla \zeta^T( {\bf x}'), 1). $

    Notice that $ |J_G| $ is bounded from above and below by two constants since $ {\bf v}\in \Sigma\cap\mathbb{S}^{d-1} $ and the Lipschitz constant of $ \zeta $ is not bigger than $ 1/2 $. Also notice that $ G(B_r'\times (0, 1)) \subset \Gamma_{r} + \Sigma\cap B_{r} $. Therefore,

    $ Brr0|(u(ξ+sv+tv)u(ξ+sv))v|ptpdsdxCΓr+ΣBr|(u(y+tv)u(y))v|ptpdyCΩB3r|(u(y+tv)u(y))v|ptpdy, $

    where we have used (4.1) in the last step. By some straightforward calculations, one can also find that $ \Omega\cap B_{r/4} \subset G(B_r'\times (0, 1)) $. Then

    $ \int_{\Omega\cap B_{\frac{r}{4}}}|{\bf u}( {\bf y})\cdot {\bf v}|^{p}d {\bf y} \leq \int_{G(B_r'\times (0, 1))}|{\bf u}( {\bf y})\cdot {\bf v}|^{p}d {\bf y} \leq C \int_{B'_{r}}\int_{0}^{r}|{\bf u}({ \boldsymbol \xi}+ s{\bf v})\cdot {\bf v}|^{p}\, ds\, d {\bf x}'. $

    It then follows from the above calculations that that for all $ {\bf v}\in \Sigma\cap\mathbb{S}^{d-1} $ and all $ t\in (0, r), $

    $ ΩBr4|u(y)v|pdyCrpΩB3r|(u(y+tv)u(y))v|ptpdy. $ (4.2)

    Multiplying the left hand side of (4.2) by $ \rho(t{\bf v})t^{d-1} $ and integrating in $ t\in(0, r) $ and in $ {\bf v}\in \Sigma\cap \mathbb{S}^{d-1} $, we get

    $ r0ΣSd1ΩBr4|u(y)v|pρ(tv)td1dydσ(v)dt=ΩBr4ΣBr|u(y)z|z||pρ(z)dzdy. $

    Using Lemma 3.4, we observe that

    $ ΩBr4ΣBr|u(y)z|z||pρ(z)dzdy=ΩBr4|u(y)|pΣBr|u(y)|u(y)|z|z||pρ(z)dzdy(r0td1ρ(t)dt)ΩBr4|u(y)|pΣSd1|u(y)|u(y)|w|pdHd1(w)dyc0(Brρ(ξ)dξ)ΩBr4|u(y)|pdy. $ (4.3)

    Similarly, we have

    $ r0ΣSd1ΩB3r|(u(y+tv)u(y))v|ptpρ(tv)td1dydσ(v)dt=ΩB3rΣBr|(u(y+z)u(y))z|z||p|z|pρ(|z|)dzdyΩB4rΩB4r|D(u)(x,y)|pρ(xy)dydx. $ (4.4)

    Combining inequalities (4.2)–(4.4) we obtain that

    $ c0ΩBr4|u(y)|pdyrpBrρ(ξ)dξΩB4rΩB4r|D(u)(x,y)|pρ(xy)dydx $ (4.5)

    for some positive constant $ c_{0} $ which only depends on $ d, p $ and the Lipschitz constant of the domain. In particular, the estimate (4.5) holds true at all boundary points $ {\overline{ \boldsymbol \eta}} \in \partial \Omega $.

    The next argument is used in the proof of [28, Lemma 5.1]. By applying standard covering argument, it follows from the inequality (4.5) that there exist positive constants $ \epsilon_{0}\in (0, 1/(2\sqrt{10})) $ and $ C $ with the property that for all $ r\in(0, r_{0}), $ such that for all $ {\bf u}\in L^{p}(\Omega; \mathbb{R}^{d}) $ that vanishes in $ \Omega_{{r}/{\sqrt{10}}} $

    $ ΩΩ2ϵ0r|u|pdxCrpBrρ(ξ)dξΩΩ|D(u)(x,y)|pρ(xy)dydx. $ (4.6)

    The positive constants $ \epsilon_{0} $ and $ C $ depend only on $ p $ and the Lipschitz character of the boundary of $ \Omega. $ For ease of calculation, set $ \tilde{r} = 2r/\sqrt{10} $. Then $ \tilde{r}/2 = r/\sqrt{10} $.

    Now let $ {\bf u}\in L^{p}(\Omega; \mathbb{R}^{d}), $ and let $ \phi\in C^{\infty}(\Omega) $ be such that: $ \phi({\bf x}) = 0 $, if $ {\bf x}\in \Omega_{\tilde{r}/2} $; $ 0\leq \phi({\bf x})\leq 1, $ if $ {\bf x}\in \Omega_{\tilde{r}/4}\setminus \Omega_{\tilde{r}/2} $; $ \phi({\bf x}) = 1, $ if $ {\bf x}\in \Omega\setminus \Omega_{\tilde{r}/4} $ and $ |\nabla \phi|\leq C/{r} $ on $ \Omega. $ Applying (4.6) to the vector field $ \phi({\bf x}){\bf u}({\bf x}) $, we obtain that

    $ ΩΩϵ0r|u|pdxCrpBrρ(ξ)dξΩΩ|D(ϕu)(x,y)|pρ(xy)dydx. $

    We may rewrite $ {\mathscr D}(\phi{\bf u}) $ as follows

    $ {\mathscr D}(\phi{\bf u})( {\bf x}, {\bf y}) = (\phi( {\bf x}) + \phi( {\bf y})) {\mathscr D}({\bf u})( {\bf x}, {\bf y}) - \left({\phi( {\bf x}){\bf u}( {\bf y}) - \phi( {\bf y}){\bf u}( {\bf x})\over | {\bf y}- {\bf x}|}\right)\cdot {( {\bf y}- {\bf x})\over | {\bf y}- {\bf x}|}. $

    It then follows that

    $ ΩΩ|D(ϕu)(x,y)|pρ(xy)dydx2p1ΩΩ|[ϕ(x)+ϕ(y)]D(u)(x,y)|pρ(xy)dydx+2p1ΩΩ|ϕ(x)u(y)ϕ(y)u(x)|yx|(yx)|yx||pρ(xy)dydx=2p1(I1+I2). $

    The first term $ I_{1} $ can be easily estimated as

    $ I1=ΩΩ|[ϕ(x)+ϕ(y)]D(u)(x,y)|pρ(xy)dydx2ΩΩ|D(u)(x,y)|pρ(xy)dydx. $

    Let us estimate the second term, $ I_{2} $. We first break it into three integrals.

    $ I2=ΩΩ|ϕ(x)u(y)ϕ(y)u(x)|yx|(yx)|yx||pρ(xy)dydx= $

    where $ A = \Omega\setminus \Omega_{\tilde{r}/4} \times \Omega\setminus \Omega_{\tilde{r}/4} $, $ B = (\Omega\setminus \Omega_{\tilde{r}/8})\times \Omega_{\tilde{r}/4} \cup \left(\Omega_{\tilde{r}/4} \times (\Omega\setminus \Omega_{\tilde{r}/8})\right) $ and $ C = \Omega\times\Omega \setminus (A\cup B) $. We estimate each of these integrals. Let us begin with the simple one: $ \iint_{A} $. After observing that $ \phi({\bf x}) = \phi({\bf y}) = 1 $ for all $ {\bf x}, {\bf y}\in \Omega\setminus \Omega_{\tilde{r}/4}, $ we have that

    $ \iint_{A} = \int_{\Omega\setminus \Omega_{\tilde{r}/4}}\int_{\Omega\setminus \Omega_{\tilde{r}/4}} | {\mathscr D}({\bf u})( {\bf x}, {\bf y})|^p\rho( {\bf x}- {\bf y})d {\bf y}\, d{\bf x}, $

    and the latter is bounded by the semi norm. Next, we note that set $ B $ is symmetric with respect to the diagonal, and as a result,

    $ \iint_{B} = 2 \int_{\Omega\setminus \Omega_{\tilde{r}/8}} \int_{\Omega_{\tilde{r}/4}} $

    and when $ ({\bf x}, {\bf y}) \in (\Omega\setminus \Omega_{\tilde{r}/8})\times \Omega_{\tilde{r}/4} $, we have $ \phi({\bf x}) = 1 $, and so we have

    $ \begin{aligned} \iint_{B} & = 2\int_{\Omega\setminus \Omega_{\tilde{r}/8}} \int_{\Omega_{\tilde{r}/4}} \left|{{\bf u}( {\bf y}) - \phi( {\bf y}){\bf u}( {\bf x})\over | {\bf y}- {\bf x}|}\cdot {( {\bf y}- {\bf x})\over | {\bf y}- {\bf x}|}\right|^p\rho( {\bf x}- {\bf y})d {\bf y}\, d{\bf x}\\ &\leq 2^{p}\int_{\Omega\setminus \Omega_{\tilde{r}/8}} \int_{\Omega_{\tilde{r}/4}} \left|\phi( {\bf y}) {\mathscr D}({\bf u})( {\bf x}, {\bf y})\right|^p\rho( {\bf x}- {\bf y})d {\bf y}\, d{\bf x}\\ &\quad + 2^{p}\int_{\Omega\setminus \Omega_{\tilde{r}/8}} \int_{\Omega_{\tilde{r}/4}} \left|{{\bf u}( {\bf y})\over | {\bf y}- {\bf x}|}\right|^p\rho( {\bf x}- {\bf y})d {\bf y}\, d{\bf x}\\ &\leq 2^{p}\int_{\Omega\setminus \Omega_{\tilde{r}/8}} \int_{\Omega_{\tilde{r}/4}}\left| {\mathscr D}({\bf u})( {\bf x}, {\bf y})\right|^p\rho( {\bf x}- {\bf y})d {\bf y}\, d{\bf x}\\ &\quad + \frac{2^{4p}}{r^{p}}\int_{\Omega\setminus \Omega_{\tilde{r}/8}} \int_{\Omega_{\tilde{r}/4}} |{\bf u}( {\bf y})|^p\rho( {\bf x}- {\bf y})d {\bf y}\, d{\bf x}\\ \end{aligned} $

    where we have used the fact that $ \text{dist}(\Omega\setminus \Omega_{\tilde{r}/8}, \Omega_{\tilde{r}/4}) = \tilde{r}/{8} $. As a consequence we have that

    $ \iint_{B} \leq 2^{p} \int_{\Omega}\int_{\Omega} | {\mathscr D}({\bf u})( {\bf x}, {\bf y})|^p \rho( {\bf y}- {\bf x})d {\bf y}\, d {\bf x} + \frac{2^{4p}}{r^{p}} \left(\int_{|{\bf h}| > \tilde{r}/{8}} \rho({\bf h}) d{\bf h} \right) \int_{\Omega_{\tilde{r}/4}} |{\bf u}( {\bf y})|^{p} d {\bf y}. $

    To estimate the integral on $ C $, we first observe that for any $ ({\bf x}, {\bf y}) \in C $, then $ \text{dist}({\bf x}, \partial \Omega)\geq \tilde{r}/{8} $ and $ \text{dist}({\bf y}, \partial \Omega)\geq \tilde{r}/{8} $. Using this information, adding and subtracting $ \phi({\bf x}){\bf u}({\bf x}) $ we can then estimate as follows:

    $ \begin{aligned} \iint_{C} &\leq 2^{p-1} \iint_{C} | {\mathscr D}({\bf u})( {\bf x}, {\bf y})|^p \rho( {\bf y}- {\bf x})d {\bf y}\, d {\bf x} \\ &\quad + 2^{p-1}\iint_{C}|{\bf u}( {\bf x})|^{p} \frac{|\phi( {\bf x} )-\phi( {\bf y})|^{p}}{| {\bf x}- {\bf y}|^{p}}\rho(| {\bf x}- {\bf y}|)d {\bf y}\, d{\bf x} \\ & \leq 2^{p-1} \int_{\Omega}\int_{\Omega} | {\mathscr D}({\bf u})( {\bf x}, {\bf y})|^p \rho( {\bf y}- {\bf x})d {\bf y}\, d {\bf x}\\ & \quad + \frac{C}{r^{p}} \int_{B_{R}} \rho({\bf h}) d{\bf h} \int_{\Omega_{{r\over8}}}|{\bf u}( {\bf x})|^{p} d {\bf x} \end{aligned} $

    where we used the estimate $ |\nabla \phi| \leq {C\over r} $, and denoted $ R = \text{diam}(\Omega) $.

    We then conclude that there exists a universal constant $ C > 0 $ such that for any $ r $ small

    $ \begin{aligned} \int_{\Omega\setminus \Omega_{\epsilon_{0} \, r }} |{\bf u}|^{p}d {\bf x} & \leq C \left({r^{p}\over \int_{B_{r}}\rho(|{ {\bf y}}|)d {\bf y}} \int_{\Omega}\int_{\Omega} | {\mathscr D}({\bf u})( {\bf x}, {\bf y})|^p \rho( {\bf y}- {\bf x})d {\bf y}\, d {\bf x} \right.\\ &\qquad +\left. \frac{1}{r^{p}}\int_{B_{R}} \rho({\bf h}) d{\bf h} \int_{\Omega_{\tilde{r}/{8}}} |{\bf u}|^{p} d {\bf x}\right) . \end{aligned} $

    It then follows that

    $ \begin{equation*} \begin{aligned} \int_{\Omega}|{\bf u}|^{p}d {\bf x} & = \int_{\Omega_{\epsilon_{0}r}}|{\bf u}|^{p}d {\bf x} + \int_{\Omega\setminus \Omega_{\epsilon_{0}r}}|{\bf u}|^{p}d {\bf x} \\ &\leq \int_{\Omega_{\epsilon_{0} r}}|{\bf u}|^{p}d {\bf x} + C\, {r^{p}\over \int_{B_{r}}\rho(|{ {\bf y}}|)d {\bf y}} \int_{\Omega}\int_{\Omega} | {\mathscr D}({\bf u})( {\bf x}, {\bf y})|^p \rho( {\bf y}- {\bf x})d {\bf y}\, d {\bf x} \\ &\quad+ C {\|\rho\|_{L^{1}(B_{R})}\over r^{p}}\int_{\Omega_{\tilde{r}/8}}|{\bf u}|^{p}d {\bf x}. \end{aligned} \end{equation*} $

    We hence complete the proof of Lemma 4.2 after choosing $ \epsilon_{0} $ sufficiently small, say for example $ \epsilon_{0} < {1}/{4\sqrt{10}} $, that

    $ \begin{equation*} \int_{\Omega}|{\bf u}|^{p}d {\bf x} \leq C(r)\int_{\Omega_{\epsilon_{0}r}}|{\bf u}|^{p}d {\bf x} + C\, {r^{p}\over \int_{B_{r}(0)}\rho(|{ {\bf y}}|)d {\bf y}} \int_{\Omega}\int_{\Omega} | {\mathscr D}({\bf u})( {\bf x}, {\bf y})|^p \rho( {\bf y}- {\bf x})d {\bf y}\, d {\bf x}, \end{equation*} $

    as desired.

    Let $ {\bf u}_{n} $ be a bounded sequence in $ \mathcal{S}_{\rho, p} (\Omega) $. Let $ \phi_{j} \in C^{\infty}_{0}(\Omega) $ such that $ \phi_{j} \equiv 1 $ in $ \Omega_{1/j} $. Then the sequence $ \{\phi_{j}{\bf u}_{n}\}_{n} $ is bounded in $ \mathcal{S}_{\rho, p} (\mathbb{R}^{d}) $, and so by Theorem 3.1, $ \phi_{j}{\bf u}_{n} $ is precompact in $ \Omega $. In particular, $ \{{\bf u}_{n}\} $ is relatively compact in $ L^{p}(\Omega_{j}) $. From this one can extract a subsequence $ {\bf u}_{n_{j}} $ such that $ {\bf u}_{n_{j}} \to {\bf u} $ in $ L^{p}_{loc}(\Omega) $. It is easy to see that $ {\bf u}\in L^{p}(\Omega) $. In fact, using the pointwise convergence and Fatou's lemma, we can see that $ {\bf u}\in \mathcal{S}_{\rho, p}(\Omega) $. What remains is to show that $ {{\bf u}_{n_{j}}} \to {\bf u} $ in $ L^{p}(\Omega) $. To that end, we apply Lemma 4.2 for the function $ {\bf u}_{n_{j}} - {\bf u} $, to obtain that

    $ \int_{\Omega}|{\bf u}_{n_{j}} - {\bf u}|^{p}d {\bf x} \leq C_{1}(r) \int_{\Omega_{\epsilon_{0}r}}|{\bf u}_{n_{j}} - {\bf u}|d {\bf x} + C_{2} \frac{r^{p}}{ \int_{B_{r}}\rho({\bf h}) d{\bf h} } |{\bf u}_{n_{j}} - {\bf u}|^{p}_{\mathcal{S}_{\rho, p}(\Omega)} $

    for all small $ r $. We now fix $ r $ and let $ j\to \infty $ to obtain that

    $ \limsup\limits_{j\to \infty}\int_{\Omega}|{\bf u}_{n_{j}} - {\bf u}|^{p}d {\bf x} \leq C \frac{r^{p}}{ \int_{B_{r}}\rho({\bf h}) d{\bf h} } (1 + |{\bf u}|^{p}_{\mathcal{S}_{\rho, p}}). $

    We then let $ r\to 0 $, to obtain that $ \limsup_{j\to \infty}\int_{\Omega}|{\bf u}_{n_{j}} - {\bf u}|^{p}d {\bf x} = 0. $

    Arguing as above and by Proposition 3.5, we have that there is a subsequence $ {\bf u}_{n_{j}} \to u $ in $ L^{p}_{loc}(\Omega) $, and that $ u\in \mathcal{S}_{\rho, p}(\Omega) $. To conclude, we apply Lemma 4.2 for the function $ {\bf u}_{n_{j}} - {\bf u} $ corresponding to $ \rho_{n_{j}} $ to obtain

    $ \int_{\Omega}|{\bf u}_{n_{j}} - {\bf u}|^{p}d {\bf x} \leq C_{1}(r) \int_{\Omega_{\epsilon_{0}r}}|{\bf u}_{n_{j}} - {\bf u}|d {\bf x} + C_{2} \frac{r^{p}}{ \int_{B_{r}}\rho_{n_{j}}({\bf h}) d{\bf h} } |{\bf u}_{n_{j}} - {\bf u}|^{p}_{\mathcal{S}_{\rho_{n_j}, p}(\Omega)} . $

    By assumption $ \rho_{n_j}\leq C \rho $ and so $ |{\bf u}_{n_{j}} - {\bf u}|^{p}_{\mathcal{S}_{\rho_{n_j}, p}(\Omega)}\leq C |{\bf u}_{n_{j}} - {\bf u}|^{p}_{\mathcal{S}_{\rho, p}(\Omega)} $. We then let $ j\to \infty $ and apply the weak convergence of $ \rho_{n} $ to obtain that

    $ \limsup\limits_{j\to \infty}\int_{\Omega}|{\bf u}_{n_{j}} - {\bf u}|^{p}d {\bf x} \leq C \frac{r^{p}}{ \int_{B_{r}}\rho({\bf h}) d{\bf h} } (1 + |{\bf u}|^{p}_{\mathcal{S}_{\rho, p}}). $

    Finally, we let $ r\to 0 $ to conclude the proof.

    We recall that given $ V\subset L^{p}(\Omega; \mathbb{R}^{d}) $ satisfying the hypothesis of the corollary, there exists a constant $ P_{0} $ such that for any $ {\bf u}\in V, $

    $ \begin{equation} \int_{\Omega} |{\bf u}|^{p}d {\bf x} \leq P_{0} \int_{\Omega}\int_{\Omega} \rho( {\bf y} - {\bf x})\left|\frac{({\bf u}( {\bf y}) - {\bf u}({ {\bf x}}))}{| {\bf y}- {\bf x}|} \cdot \frac{( {\bf y} - {\bf x})}{| {\bf y} - {\bf x}|}\right|^{p}d {\bf y} d {\bf x}. \end{equation} $ (4.7)

    This result is proved in [12] or [23]. We take $ P_{0} $ to be the best constant. We claim that given any $ \epsilon > 0 $, there exists $ N = N(\epsilon) \in \mathbb{N} $ such that for all $ n\geq N $, (1.15) holds for $ C = P_{0} + \epsilon $. We prove this by contradiction. Assume otherwise and that there exists $ C > P_{0} $ such that for every $ n $, there exists $ {\bf u}_{n}\in V\cap L^{p}(\Omega; \mathbb{R}^{d}) $, $ \|{\bf u}_{n}\|_{L^{p}} = 1 $, and

    $ \int_{\Omega}\int_{\Omega} \rho_{n}( {\bf y} - {\bf x})\left|\frac{({\bf u}_{n}( {\bf y}) - {\bf u}_{n}({ {\bf x}}))}{| {\bf y}- {\bf x}|} \cdot \frac{( {\bf y} - {\bf x})}{| {\bf y} - {\bf x}|}\right|^{p}d {\bf y} d {\bf x} < \frac{1}{C}. $

    By Theorem 1.4, $ {\bf u}_{n} $ is precompact in $ L^{p}(\Omega; \mathbb{R}^{d}) $ and therefore any limit point $ {\bf u} $ will have $ \|{\bf u}\|_{L^{p}} = 1 $, and will be in $ V\cap L^{p}(\Omega; \mathbb{R}^{d}) $. Moreover, following the same procedure as in the proof of Proposition 3.5, we obtain that

    $ \begin{aligned} \int_{\Omega}\int_{\Omega} &\rho( {\bf y} - {\bf x})\left|\frac{({\bf u}( {\bf y}) - {\bf u}({ {\bf x}}))}{| {\bf y}- {\bf x}|} \cdot \frac{( {\bf y} - {\bf x})}{| {\bf y} - {\bf x}|}\right|^{p}d {\bf y} d {\bf x} \\ &\leq \liminf\limits_{n\to \infty}\int_{\Omega}\int_{\Omega} \rho_{n}( {\bf y} - {\bf x})\left|\frac{({\bf u}_{n}( {\bf y}) - {\bf u}_{n}({ {\bf x}}))}{| {\bf y}- {\bf x}|} \cdot \frac{( {\bf y} - {\bf x})}{| {\bf y} - {\bf x}|}\right|^{p}d {\bf y} d {\bf x} \leq {1 \over C } < {1\over P_{0}} \end{aligned} $

    which gives the desired contradiction since $ P_{0} $ is the best constant in (4.7).

    In this work, we have presented a set of sufficient conditions that guarantee a compact inclusion of a set of $ L^{p} $-vector fields in the Banach space of $ L^{p} $ vector fields. The criteria are nonlocal and given with respect to nonlocal interaction kernels that may not be necessarily radially symmetric. We note that, in addition to the mathematical generality, relaxing the radial symmetry assumption on nonlocal interactions can be useful when modeling anisotropic behavior and directional transport. The $ L^{p} $-compactness is established for a sequence of vector fields where the nonlocal interactions involve only part of their components, so that the results and discussions represent a significant departure from those known for scalar fields. It is not clear yet to what extent the conditions assumed here can be weakened to reach the same conclusions. In this regard, there are still some outstanding questions in relation to the set of minimal conditions on the interaction kernel as well as on the set of vector fields that imply $ L^{p} $-compactness. An application of the compactness result that will be explored elsewhere includes designing approximation schemes for nonlocal system of equations of peridynamic-type similar to the one done in [34] for nonlocal equations.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    Q. Du's research is partially supported by US National Science Foundation grant DMS-1937254 and DMS-2309245.

    T. Mengesha's research is supported by US National Science Foundation grant DMS-1910180 and DMS-2206252.

    X. Tian's research is supported by US National Science Foundation grant DMS-2111608 and DMS-2240180.

    The authors thank Zhaolong Han for helping improve the proof of Lemma 4.2.

    The authors declare no conflicts of interest.

    Compactness in $ L^{p}_{loc} $ topology

    The following theorem as well as the proof we present here is inspired by the compactness result proved in [18] (see also [17]) for scalar functions which uses the more flexible nonintegrability condition of the $ {\rho({\bf z}) \over |{\bf z}|^{p}} $ than the one stated in (1.11).

    Theorem A.1 ($ L^{p}_{loc} $ compactness). Suppose that $ 1\leq p < \infty $. Let $ \rho \in L^{1}(\mathbb{R}^{d}) $ be a nonnegative radial function satisfying

    $ \begin{equation} \lim\limits_{\delta\to 0 } \int_{|{\bf z}| > \delta} {\rho({\bf z})\over |{\bf z}|^{p}}d{\bf z} = \infty. \end{equation} $ (A.1)

    Suppose also that $ \{{\bf u}_{n}\} $ is a sequence of vector fields that is bounded in $ \mathcal{S}_{\rho, p}(\mathbb{R}^{d}) $. Then for any $ D\subset \mathbb{R}^{d} $ open and bounded, the sequence $ \{{\bf u}_{n}|_{D}\} $ is precompact in $ L^{p}(D; \mathbb{R}^{d}) $.

    As stated earlier, for radial functions with compact support, condition (9.1) is weaker than (1.11). Indeed, (1.11) implies that $ \rho({\bf z}) | {\bf z}|^{-p} $ is not integrable near $ { \bf 0} $ which implies (9.1). One the other hand, the kernel $ \rho({\bf z}) = |{\bf z}|^{-d-p} \chi_{B_{1}({ \bf 0})}({\bf z}) $ satisfies (A.1) but not (1.11).

    Similar to the argument we gave in Section 2, the proof of the theorem will make use of the following variant of the Riesz-Fréchet-Kolomogorov theorem [6,20].

    Lemma A.2. ([20, Lemma 5.4]) Let the sequence $ \{\mathbb{G}^\delta\}_{\delta > 0}\subset L^1(\mathbb{R}^{d}; \mathbb{R}^{d\times d}) $ be an approximation to the identity. That is

    $ \forall \delta > 0, \int_{\mathbb{R}^{d}} \mathbb{G}^\delta ( {\bf x})d {\bf x} = \mathbb{I}_d, \quad \mathit{\text{for any $r > 0$, }} \lim\limits_{\delta \to 0}\int_{| {\bf x}| > r} \mathbb{G}^\delta ( {\bf x})d {\bf x} = {\bf 0}. $

    If $ \{{\bf f}_n\}_{n} $ is a bounded sequence in $ L^{p}(\mathbb{R}^{d}; \mathbb{R}^{d}) $ and

    $ \lim\limits_{\delta \to 0} \limsup\limits_{n\to \infty} \|{\bf f}_{n} - \mathbb{G}^\delta \ast {\bf f}_{n}\|_{L^{p}} = 0, $

    then for any open and bounded subset $ D $ of $ \mathbb{R}^{d} $ the sequence $ \{{\bf f}_{n}\} $ is relatively compact in $ L^{p}(D; \mathbb{R}^{d}) $.

    Proof of Theorem A.1. From the assumption we have

    $ \begin{equation} \sup\limits_{n\geq 1} \|{\bf u}_{n}\|_{L^{p}}^{p} + \sup\limits_{n\geq 1}\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}\rho( {\bf x}'- {\bf x})\left| {\mathscr D}({\bf u}_n)( {\bf x}, {\bf x}')\right|^{p}d {\bf x}'d {\bf x} < \infty. \end{equation} $ (A.2)

    Let $ \Gamma^{\delta}({\bf z}) = {\rho({\bf z}) \over |{\bf z}|^{p}} \chi_{\complement B_{\delta}}({\bf z}) $. Then for each $ \delta $, $ \Gamma^\delta \in L^1(\mathbb{R}^{d}) $ and is radial, since $ \rho $ is radial. Moreover, by assumption on $ \rho $ (A.1), $ \|\Gamma^\delta\|_{L^1} \to \infty $ as $ \delta \to 0. $ We next introduce the following sequence of integrable matrix functions

    $ \mathbb{G}^{\delta}({\bf z}) = {d \, \Gamma^{\delta}({\bf z})\over \|\Gamma^\delta\|_{L^1} } {{\bf z}\otimes {\bf z} \over |{\bf z}|^{2}}. $

    Notice that since $ \Gamma^\delta $ is radial, we have

    $ \int_{\mathbb{R}^{\delta}} {\Gamma^{\delta}({\bf z}) z_i^{2} \over |{\bf z}|^{2}}d{\bf z} = { \|\Gamma^\delta\|_{L^1}\over d}, \quad i = 1, \cdots, d. $

    As a consequence $ \{\mathbb{G}^{\delta}\} $ is an approximation to the identity. Now for each $ n $ we have

    $ \begin{aligned} \| {\bf u}_{n} - \mathbb{G}^{\delta} \ast{\bf u}_n\|_{L^{p}}^{p} & = \int_{\mathbb{R}^{d}}\left|\int_{\mathbb{R}^{d}}\mathbb{G}^{d} ( {\bf y}- {\bf x})({\bf u}_n( {\bf x}) - {\bf u}_n( {\bf y})) d {\bf y} \right|^{p} d {\bf x}\\ & \leq d^{p} \int_{\mathbb{R}^{d}}\left|\int_{\mathbb{R}^{d}} {\Gamma^{\delta}({\bf z})\over \|\Gamma^\delta\|_{L^1} }\left|{ {\bf z}\over | {\bf z}|}\cdot({\bf u}_n( {\bf x}) - {\bf u}_n( {\bf z} + {\bf x})) \right|d {\bf y} \right|^{p} d {\bf x}\\ &\leq d^{p} \int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}} \left|{ {\bf z}\over | {\bf z}|}\cdot({\bf u}_n( {\bf x}) - {\bf u}_n( {\bf z} + {\bf x})) \right|^{p} {\Gamma^{\delta}({\bf z})\over \|\Gamma^\delta\|_{L^1} } d {\bf z} d {\bf x}\\ &\leq {d^{p}\over \|\Gamma^\delta\|_{L^1} } \int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}} \left|{ {\bf z}\over | {\bf z}|}\cdot({\bf u}_n( {\bf x}) - {\bf u}_n( {\bf z} + {\bf x})) \right|^{p} {\rho({\bf z}) \over |{\bf z}|^{p}} d {\bf z} d {\bf x}\\ & = {d^{p}\over \|\Gamma^\delta\|_{L^1} }|{\bf u}_{n}|^p_{\mathcal{S}_{\rho, p}(\mathbb{R}^{d})}. \end{aligned} $

    By assumption on the sequence $ \{{\bf u}_n\} $ (A.2), we have that for all $ n $,

    $ \| {\bf u}_{n} - \mathbb{G}^{\delta} \ast{\bf u}_n\|_{L^{p}}^{p} \leq C \, {d^{p}\over \|\Gamma^\delta\|_{L^1}}. $

    We take the limit as $ \delta\to 0 $ (uniformly in $ n $) and use Lemma A.2 to conclude that $ {\bf u}_n $ is compact in $ L^{p}(\Omega; \mathbb{R}^{d}). $

    [1] Stoeck K, Schmitz M, Ebert E, et al. (2014) Immune responses in rapidly progressive dementia: a comparative study of neuroinflammatory markers in Creutzfeldt-Jakob disease, Alzheimer's disease and multiple sclerosis. J Neuroinflam 11: 170-178. doi: 10.1186/s12974-014-0170-y
    [2] Kojima G, Tatsuno BK, Inaba M, et al. (2013) Creutzfeldt-Jakob disease: a case report and differential diagnosis. Hawai'i J Med Public Health 72: 136-139.
    [3] Manuelidis EE, Manuelidis L (1989) A clinical series with 13% of Alzheimer's disease actually CJD. Alz Dis Assoc Disorders 3: 100-109. doi: 10.1097/00002093-198903010-00009
    [4] Bastian FO, McDermont ME, Perry AS, et al. (2005) Safe method for isolation of prion protein and diagnosis of Creutzfeldt-Jakob disease. J Virol Methods 130: 133-139. doi: 10.1016/j.jviromet.2005.06.024
    [5] Nicolson GI (2008) Chronic bacterial and viral infections in neurodegenerative and neurobehavioral diseases. Lab Med 39: 291-299. doi: 10.1309/96M3BWYP42L11BFU
    [6] Bastian FO (1991) Author, Creutzfeldt-Jakob Disease and Other Transmissible Spongiform Encephalopathies, New York, Mosby/Year Book 256 pp
    [7] Matthews WB (1978) Creutzfeldt-Jakob disease. Postgrad Med J 54: 591-594. doi: 10.1136/pgmj.54.635.591
    [8] Manuelidis EE, De Figuriredo JM, Kim JH, et al. (1988) Transmission studies from blood of Azheimer disease patients and healthy relatives. Proc Natl Acad Sci (USA) 85: 4898-4901. doi: 10.1073/pnas.85.13.4898
    [9] Manuelidis EE, Gorgacz EJ, Manuelidis L (1978) Transmission of Creutzfeldt-Jakob disease with scrapie-like syndromes to mice. Nature 271: 778-779.
    [10] Hainfellner JA, Wanschitz J, Jellinger K, et al. (1998) Coexistence of Alzheimer-type neuropathology in Creutzfeldt_Jakob disease. Acta Neuropathol 96: 116-122. doi: 10.1007/s004010050870
    [11] Solito E, Sastre M (2012) Microglia function in Alzheimer's disease. Frontiers Pharmacol 3: 1-10.
    [12] Tousseyn T, Bajsarowicz K, Sanchez H, et al. (2015) Prion disease induces Alzheimer disease-like neuropathologic changes. J Neuropathol Exp Neurol 74: 873-888.
    [13] Dheen ST, Kaur C, Ling EA (2007) Microglial activation and its implications in the brain diseases. Current Med Chem 14: 1189-1197. doi: 10.2174/092986707780597961
    [14] Mattson MP (2002) Oxidative stress, perturbed calcium homeostasis, and immune dysfunction in Alzheimer's disease. J Neurovirol 8: 539-550. doi: 10.1080/13550280290100978
    [15] Garcao P, Oliveira CR, Agostinho P (2006) Comparative study of microglia activation induced by amyloid-β and prion peptides. J Neurosci Res 84: 182-193. doi: 10.1002/jnr.20870
    [16] Lee DY, Lee J, Sugden B (2009) The unfolded protein response and autophagy: herpes viruses rule! J Virol 83: 1168-1172.
    [17] Unterberger U, Hoftberger R, Gelpi E, et al. (2006) Endoplasmic reticulum stress features are prominent in Alzheimer's disease but not in prion disease in vivo. J Neuropathol Exp Neurol 65: 348-357. doi: 10.1097/01.jnen.0000218445.30535.6f
    [18] Sorce S, Nuvolone M, Keller A, et al. (2014) The role of NADPH oxidase NOX2 in prion pathogenesis. PLoS Pathogens 10: e1004531.
    [19] Bastian FO (2014) Cross-roads in research on neurodegenerative diseases. J Alzheimer's Dis Parkinsonism 4: 141. doi:10.4172/21610460.1000141.
    [20] Moreno JA, Radford H, Peretti D, et al. (2012) Sustained translational repression of eIF2α mediates prion neurodegeneration. Nature 485: 507-511.
    [21] Greenlee JJ, Greenlee MH (2015) The transmissible spongiform encephalopathies of livestock. Ilar J 56: 7-25. doi: 10.1093/ilar/ilv008
    [22] Baker CA, Martin D, Manuelidis L (2002) Microglia from Creutzfeldt-Jakob disease-infected brains are infectious and show specific mRNA activation profiles. J Virol 76: 10905-10913. doi: 10.1128/JVI.76.21.10905-10913.2002
    [23] Murali A, Maue RA, Dolph PJ (2014) Reversible symptoms and clearance of mutant prion protein in an inducible model of a genetic prion disease in Drosophilia melanogaster. Neurobiol Dis 67: 71-78. doi: 10.1016/j.nbd.2014.03.013
    [24] Sala I, Marquie M, Sanchez-Saudinos MB, et al. (2012) Rapidy progressive dementia: experience in a tertiary care medical center. Alzheimer Dis Assoc Disorders 26: 267-271. doi: 10.1097/WAD.0b013e3182368ed4
    [25] Armitage WJ, Tullo AB, Ironside JW (2009) Risk of Creutzfeldt-Jakob disease transmission by ocular surgery and tissue transplantation. Eye 23: 1926-1930. doi: 10.1038/eye.2008.381
    [26] Wemheuer WM, Benestad SL, Wrede A, et al. (2009) Similarities between forms of sheep scrapie and Creutzfeldt-Jakob disease are encoded by distinct prion types. Amer J Pathol 175: 2566-2573. doi: 10.2353/ajpath.2009.090623
    [27] Cassard H, Torres JM, Lacroux C, et al. (2014) Evidence for zoonotic potential of ovine scrapie prions. Nature communications 5: 5821 doi:10.1038/ncomms6821.
    [28] Merz PA, Somerville RA, Wisniewski HM, et al. (1983) Scrapie-associated fibrils in Creutzfeldt-Jakob disease. Nature 306: 474-476.
    [29] Wisniewski T, Aucouturier P, Soto C, et al. (1998) The prionoses and other conformational disorders. Amyloid 5: 212-224. doi: 10.3109/13506129809003848
    [30] Serrano-Pozo A, Frosch MP, Masliah E, et al. (2011) Neuropathological alterations in Alzheimer Disease. Cold Spring Harb Perspect Med 1: a006189.
    [31] Kessels HW, Nguyen LN, Nabavi S, et al. (2010) The prion protein as a receptor for amyloid-β. Nature 466: 7308 E3-4.
    [32] Kumar A, Pate KM, Moss MA, et al. (2014) Self-propagative replication of A-β oligomers suggests potential transmissibility in Alzheimer disease. PLoS ONE 9: e111492. doi: 10.1371/journal.pone.0111492
    [33] Guo JL, Lee VM (2011) Seeding of normal Tau by pathological Tau conformers drives pathogenesis of Alzheimer-like tangles. J Biol Chem 286: 15317-15331.
    [34] Liu L, Drouet V, Wu JW, et al. (2012) Trans-synaptic spread of Tau pathology in vivo. PloS ONE 7: e31302. Doi:10.1371. doi: 10.1371/journal.pone.0031302
    [35] Eisele YS, Bolmont T, Heikenwalder M, et al. (2009) Induction of cerebral β-amyloidosis: intracerebral versus systemic Aβ inoculation. PNAS 106: 12926-12931. doi: 10.1073/pnas.0903200106
    [36] Morales R, Duran-Aniotz C, Castilla J, et al. (2011) De novo induction of amyloid-β deposition in vivo. Mol Psychiatry 17: 1347-1353.
    [37] Volpicelli-Daley LA, Luk KC, Patel TP, et al. (2011) Exogenous α-synuclein fibrils induce Lewy body pathology leading to synaptic dysfunction and neuron death. Neuron 72: 57-71. doi: 10.1016/j.neuron.2011.08.033
    [38] Luk KC, Kehm VM, Zhang B, et al. (2012) Intracerebral inoculation of pathological α-synuclein initiates a rapidly progressive neurodegenerative α-synucleinopathy in mice. J Exp Med 209: 975-986. doi: 10.1084/jem.20112457
    [39] Stöhr J, Watts JC, Mensinger ZL, et al. Purified and synthetic Alzheimer's amyloid β (Aβ) prions. PNAS 109: 11025-11030.
    [40] Ghoshal N, Cali I, Perrin RJ, et al. (2009) Co-distribution of amyloid β plaques and spongiform degeneration in familial Creutzfeldt-Jakob disease with the E200K-129M haplotype. Arch Neurol 66: 1240-1246.
    [41] Salvadores N, Shahnawaz M, Scarpini E, et al. (2014) Detection of misfolded Aβ oligonmers for sensitive biochemical diagnosis of Alzheimer's disease. Cell Reports 7: 261-268. doi: 10.1016/j.celrep.2014.02.031
    [42] Morales R, Moreno-Gonzalez I, Soto C (2013) Cross-seeding of misfolded proteins: implications for etiology and pathogenesis of protein misfolding diseases. PLOS 9: e1003537. doi: 10.1371/journal.pgen.1003537
    [43] Chi EY, Frey SL, Winans A, et al. (2010) Amyloid-β fibrillogenesis seeded by interface-induced peptide misfolding and self-assembly. Biophy J 98: 2299-2308. doi: 10.1016/j.bpj.2010.01.056
    [44] Tateishi J, Kitamoto T, Hoque MZ, et al. (1996) Experimental transmission of Creutzfeldt-Jakob disease and related diseases to rodents. Neurology 46: 532-537. doi: 10.1212/WNL.46.2.532
    [45] Kovacs GG, Seguin J, Quadrio I, et al. (2011) Genetic Creutzfeldt-Jakob disease associated with the E200K mutation: characterization of a complex proteinopathy. Acta Neuropathologica 121: 39-57. doi: 10.1007/s00401-010-0713-y
    [46] Vital A, Canron M-H, Gil R, et al. (2007) A sporadic case of Creutzfeldt-Jakob disease with β-amyloid deposits and α-synuclein inclusions. Neuropathology 27: 273-277. doi: 10.1111/j.1440-1789.2007.00755.x
    [47] Kasai T, Tokuda T, Ishii R, et al. (2014) Increased α-synuclein levels in the cerebrospinal fluid of patients with Creutzfeldt–Jakob disease. J Neurol 261: 7334-7337.
    [48] Zhang M, Hu R, Chen H, et al. (2015) Polymorphic cross-seeding amyloid assemblies of amyloid-β and human islet amyloid peptide. Phys Chem Chem Phys 17: 23245-23256. doi: 10.1039/C5CP03329B
    [49] O'Nuallain B, Williams AD, Westermark P, et al. (2004) Seeding specificity in amyloid growth induced by heterologous fibrils. J Biol Chem 279: 17490-17499. doi: 10.1074/jbc.M311300200
    [50] Westermark P, Westermark GT (2013) Seeding and cross-seeding in amyloid diseases, in Zucker M, Christen Y (eds.) Proteopathic Seeds and Neurodegenerative Diseases, Research and Perspectives in Alzheimer' Disease, Berlin, Springer-Verlag pp. 47-60.
    [51] Zhou Y, Smith D, Leong BJ, et al. (2012) Promiscuous cross-seeding between bacterial amyloids promotes interspecies biofilms. J Biol Chem 287: 35092-35103. doi: 10.1074/jbc.M112.383737
    [52] Prusiner SB (1987) Prions causing degenerative neurological diseases. Ann Rev Med 38: 381-398. doi: 10.1146/annurev.me.38.020187.002121
    [53] Vila-Vicosa D, Campos SR, Baptista AM, et al. (2012) Reversibility of prion misfolding: insights from constant –pH molecular dynamics simulations. J Physical Chem 116: 8812-8821. doi: 10.1021/jp3034837
    [54] Eisenberg D, Jucker M (2012) The amyloid state of proteins in human diseases. Cell 148: 1188-1203. doi: 10.1016/j.cell.2012.02.022
    [55] Mayer RJ, Landon M, Laszlo L, et al. (1992) Protein processing in lysosomes: the new therapeutic target in neurodegenerative disease. Lancet 340: 156-159. doi: 10.1016/0140-6736(92)93224-B
    [56] Rigter A, Priem J, Langeveld JP, et al. (2011) Prion protein self-interaction in prion disease therapy approaches. Vet Quarterly 31: 115-128. doi: 10.1080/01652176.2011.604976
    [57] Safar JG (2012) Molecular pathogenesis of sporadic prion diseases in man. Prion 6: 108-115. doi: 10.4161/pri.18666
    [58] Tuite MF, Cox BS (2003) Propagation of yeast prions. Nature Rev Mol Cell Biol 4: 878-890. doi: 10.1038/nrm1247
    [59] Bellinger-Kawahara C, Diener TO, McKinley MP, et al. (1987) Purified scrapie prions resist inactivation by procedures that hydrolyze, modify, or shear nucleic acids. Virology 160: 271-274. doi: 10.1016/0042-6822(87)90072-9
    [60] Hearst JE (1981) Psoralen photochemistry and nucleic acid structure. J Investigative Dermatol 77: 39-44. doi: 10.1111/1523-1747.ep12479229
    [61] Miyazawa K, Kipkorir T, Tittman S, et al. (2012) Continuous production of prions after infectious particles are eliminated: implications for Alzheimer's disease. PLoS ONE 7: 1-8.
    [62] Sun R, Liu Y, Zhang H, et al. (2008) Quantitative recovery of scrapie agent with minimal protein from highly infectious cultures. Viral Immunol 21:293-302.
    [63] Soto C, Estrada L, Castilla J (2006) Amyloids, prions and the inherent infectious nature of misfolded protein aggregates. Trends Biochem Sci 31: 150-155. doi: 10.1016/j.tibs.2006.01.002
    [64] Klingeborn M, Race B, Meade-White KD, et al. (2011) Lower specific infectivity of protease-resistant prion protein generated in cell-free reactions. PNAS 108: E1244-E1253. doi: 10.1073/pnas.1111255108
    [65] Manuelidis L (2011) Nuclease resistant circular DNAs copurify with infectivity in scrapie and CJD. J Neurovirol 17: 131-145.
    [66] Sonati T, Reimann RR, Falsig J, et al. (2013) The toxicity of antiprion antibodies is mediated by the flexible tail of the prion protein. Nature 501: 102-106.
    [67] Miyazawa K, Emmerling K, Manuelidis L (2011) Replication and spread of CJD, kuru and scrapie agents in vivo and in cell culture. Virulence 2: 188-199. doi: 10.4161/viru.2.3.15880
    [68] Watarai M, Kim S, Erdenebaatar J, et al. (2003) Cellular prion protein promotes Brucella infection into macrophages. J Exp Med 198: 5-17.
    [69] Bastian FO (2005) Spiroplasma as a candidate causal agent of transmissible spongiform encephalopathies. J Neuropathol Exp Neurol 64: 833-838.
    [70] Bastian FO (2014) The case for involvement of spiroplasma in the pathogenesis of transmissible spongiform encephalopathies. J Neuropathol Exp Neurol 73: 104-114.
    [71] Baker C, Martin D, Manuelidis L (2002) Microglia from Creutzfeldt-Jakob disease-infected brains are infectious and show specific mRNA activation profiles. J Virol 76: 10905-10913.
    [72] Marlatt MW, Bauer J, Aronica E, et al. (2014) Proliferation in the Alzheimer hippocampus is due to microglia, not astroglia, and occurs at sites of amyloid deposition. Neural Plasticity 2014: 693851 1-12.
    [73] Miklossy J, Kis A, Radenovic A, et al. (2006) Β-amyloid deposition and Alzheimer's type changes induced by Borrelia spirochetes. Neurobiol Aging 27: 228-236. doi: 10.1016/j.neurobiolaging.2005.01.018
    [74] Balin BJ, Little CS, Hammond CJ, et al. (2008) Chlamydophila pneumonia and the etiology of late-onset Alzheimer's disease. J Alzheimer's Dis 13: 371-380.
    [75] Poole S, Singhrao SK, Kesavalu L, et al. (2013) Determining the presence of peridontopathic virulence factors in short-term postmortem Alzheimer's disease brain tissue. J Alzheimer's Dis 36: 665-677.
    [76] Singhrao SK, Harding A, Poole S, et al. (2015) Porphyromonas gingivalis periodontal infection and it putative links with Alzheimer's disease. Mediators Inflam 2015: 137357.
    [77] Singhrao SK, Harding A, Simmons T, et al. (2014) Oral inflammation, tooth loss, risk factors, and association with progression of Alzheimer's disease. J Alzheimer's Dis 42: 723-737.
    [78] Miklossy J (2008) Chronic inflammation and amyloidogenesis in Alzheimer's disease- role of spirochetes. J Alzheimer's Dis 13: 381-391.
    [79] Friedland RP (2015) Mechanisms of molecular mimicry involving the microbiota in neurodegeneration. J Alzheimer's Dis 45: 349-362.
    [80] Little CS, Joyce TA, Hammond CJ, et al. (2014) Detection of bacterial antigens and Alzheimer's disease-like pathology in the central nervous system of BALB/c mice following intranasal infection with a laboratory isolate of Chlamydia pneumoniae. Frontiers Aging Neurosci 6: 304. doi: 10.3389/fnagi.2014.00304 1-9.
    [81] Goto S, Anbutsu H, Fukatsu T (2006) Asymmetrical interactions between Wolbachia and Spiroplasma endosymbionts coexisting in the same insect host. Applied Environmental Microbiol 72: 4805-4810. doi: 10.1128/AEM.00416-06
    [82] Takahashi Y, Mihara H (2004) Construction of chemically and conformationally self-replicating system of amyloid-like fibrils. Bioorg Med Chem 12: 693-699. doi: 10.1016/j.bmc.2003.11.022
    [83] Barnhart MM, Chapman MR (2006) Curli biogenesis and function. Annu Rev Microbiol 60: 131-147.
    [84] Wang X, Chapman MR (2008) Curli provide the template for understanding controlled amyloid propagation. Prion 2: 57-60. doi: 10.4161/pri.2.2.6746
    [85] Lundmark K, Westermark G, Olsen A, et al. (2005) Protein fibrils in nature can enhance amyloid protein A amyloidosis in mice: cross-seeding as a disease mechanism. PNAS 102: 6098-6102. doi: 10.1073/pnas.0501814102
    [86] Bastian FO (1979) Spiroplasma-like inclusions in Creutzfeldt-Jakob disease. Arch Pathol Lab Med 103: 665-669.
    [87] Bastian FO, Hart MN, Cancilla PA (1981) Additional evidence of spiroplasma in Creutzfeldt-Jakob disease. Lancet 1: 660.
    [88] Gray A, Francis RJ, Scholtz CL (1980) Spiroplasma and Creutzfeldt-Jakob disease. Lancet 2, 660.
    [89] Reyes JM, Hoenig EM (1981) Intracellular spiral inclusions in cerebral cell processes in Creutzfeldt-Jakob disease. J Neuropathol Exp Neurol 40: 1-8. doi: 10.1097/00005072-198101000-00001
    [90] Alexeeva I, Elliott EJ, Rollins S, et al. (2006) Absence of spiroplasma or other bacterial 16s rRNA genes in brain tissue of hamsters with scrapie. J Clin Microbiol 44: 91-97. doi: 10.1128/JCM.44.1.91-97.2006
    [91] Bastian FO, Dash S, Garry RF (2004) Linking chronic wasting disease to scrapie by comparison of Spiroplasma mirum ribosomal DNA sequences. Exp Mol Pathol 77: 49-56. doi: 10.1016/j.yexmp.2004.02.002
    [92] Bastian FO, Sanders DE, Forbes WA, et al. (2007) Spiroplasma spp. from transmissible spongiform encephalopathy brains or ticks induce spongiform encephalopathy in ruminants. J Med Microbiol 56: 1235-1242.
    [93] Bastian FO, Boudreaux CM, Hagius SD, et al. (2011) Spiroplasma found in the eyes of scrapie affected sheep. Vet Ophthalmol 14: 10-17.
    [94] Bastian FO, Purnell DM, Tully JG (1984) Neuropathology of spiroplasma infection in the rat brain. Am J Pathol 114: 496-514.
    [95] Tully JG, Bastian FO, Rose DL (1984) Localization and persistence of spiroplasmas in an experimental brain infection in suckling rats. Ann Microbiol (Paris) 135A: 111-117.
    [96] Bastian FO, Jennings R, Huff C (1987) Neurotropic Response of Spiroplasma mirum following peripheral inoculation in the rat. Ann Microbiol (Inst Pasteur) 138: 651-655. doi: 10.1016/0769-2609(87)90143-8
    [97] Jeffrey M, Scott JR, Fraser H (1991) Scrapie inoculation of mice: light and electron microscopy of the superior colliculi. Acta Neuropathol 81: 562-571.
    [98] Trachtenberg S, Gilad R (2001) A bacterial linear motor: cellular and molecular organization of the contractile cytoskeleton of the helical bacterium Spiroplasma melliferum BC3. Mol Microbiol 41: 827-848.
    [99] Bastian FO, Jennings R, Gardner W (1987) Antiserum to scrapie associated fibril protein cross-reacts with Spiroplasma mirum fibril proteins. J Clin Microbiol 25: 2430-2431.
    [100] Bastian FO, Elzer PH, Wu X (2012) Spiroplasma spp. biofilm formation is instrumental for their role in the pathogenesis of plant, insect and animal diseases. Exp Mol Pathol 93: 116-128.
    [101] Forloni G, Iussich S, Awan T, et al. (2002) Tetracyclines affect prion infectivity. PNAS 99: 10849-10854.
    [102] Guo YJ, Han J, Yao HL, et al. (2007) Treatment of scrapie pathogen 263K with tetracycline partially abolishes protease-resistant activity in vitro and reduces infectivity in vivo. Biomed Environ Sci 20: 198-202.
    [103] Haig DA, Pattison IH (1967) In-vitro growth of pieces of brain from scrapie-affected mice. J Path Bact 93: 724-727. doi: 10.1002/path.1700930243
    [104] Pattison IH (1969) Scrapie- a personal view. J Clin Pathol (Supp) 6: 110-114.
    [105] Sinclair SH, Rennoll-Bankert KE, Dumier JS (2014) Effector bottleneck: microbial reprogramming of parasitized host cell transcription by epigenetic remodeling of chromatin structure. Front Genetics 5: 274. doi: 10.3389/fgene.2014.00274 1-10.
    [106] Di Francesco A, Arosio B, Falconi A, et al. (2015) Global changes in DNA methylation in Alzheimer's disease peripheral blodd mononuclear cells. Brain Behav Immun 45: 139-144.
    [107] Drury JL, Chung WO (2015) DNA methylation differentially regulates cytokine secretion in gingival epithelia in response to bacterial challengs. Pathogens Dis 73: 1-6.
    [108] Cal H, Xie Y, Hu L, et al. (2013) Prion protein (PrPc) interacts with histone H3 confirmed by affinity chromatography. J Chromat Analytical Tech Biomed Life Sci 929: 40-44. doi: 10.1016/j.jchromb.2013.04.003
    [109] Derail M, Mill J, Lunnon K (2014) The mitochondrial epigenome: a role in Alzheimer's disease? Epigenomics 6: 665-675. doi: 10.2217/epi.14.50
    [110] Choi HS, Choi YG, Shinn HY, et al. (2014) Dysfunction of mitochondrial dynamics in the brains of scrapie-infected mice. Biochem Biophys Res Comm 448: 157-162. doi: 10.1016/j.bbrc.2014.04.069
    [111] Razin S, Yogev D, Naot Y (1998) Molecular biology and pathogenicity of mycoplasmas. Microbiol Mol Biol Rev 62: 1094-1156.
    [112] Nur I, Szyf M, Razin A, et al. (1985) Procaryotic and eukaryotic traits of DNA methylation in spiroplasmas. J Bacteriol 164: 19-24.
    [113] Halfmann R, Lindquist S (2010) Epigenetics in the extreme: prions and the inheritance of environmentally acquired traits. Science 330: 629-632. doi: 10.1126/science.1191081
    [114] Bastian FO (2014) Cross-roads in research on neurodegenerative diseases. J Alzheimer's Dis Parkinsonism 4: 1000141.
    [115] Bleme H, Hamon M, Cossart P (2012) Epigenetics and bacterial infections. Cold Spring Harb Perspect Med 2: a010272.
  • This article has been cited by:

    1. Niloofar Hassannejad, Abbas Bahador, Nasim Hayati Rudbari, Mohammad Hossein Modarressi, Kazem Parivar, In vivo antibacterial activity of Zataria multiflora Boiss extract and its components, carvacrol, and thymol, against colistin‐resistant Acinetobacter baumannii in a pneumonic BALB/c mouse model , 2019, 120, 0730-2312, 18640, 10.1002/jcb.28908
    2. Umar Ndagi, Abubakar A. Falaki, Maryam Abdullahi, Monsurat M. Lawal, Mahmoud E. Soliman, Antibiotic resistance: bioinformatics-based understanding as a functional strategy for drug design, 2020, 10, 2046-2069, 18451, 10.1039/D0RA01484B
    3. Saima Muzammil, Mohsin Khurshid, Iqra Nawaz, Muhammad Hussnain Siddique, Muhammad Zubair, Muhammad Atif Nisar, Muhammad Imran, Sumreen Hayat, Aluminium oxide nanoparticles inhibit EPS production, adhesion and biofilm formation by multidrug resistant Acinetobacter baumannii, 2020, 36, 0892-7014, 492, 10.1080/08927014.2020.1776856
    4. Kaoru YAMABE, Yukio ARAKAWA, Masaki SHOJI, Mitsuko ONDA, Katsushiro MIYAMOTO, Takahiro TSUCHIYA, Yukihiro AKEDA, Kuniko TERADA, Kazunori TOMONO, Direct anti-biofilm effects of macrolides on Acinetobacter baumannii: comprehensive and comparative demonstration by a simple assay using microtiter plate combined with peg-lid, 2020, 41, 0388-6107, 259, 10.2220/biomedres.41.259
    5. Kangjian Zhang, Xiaoyan Yang, Jiali Yang, Xia Qiao, Feng Li, Xiaoming Liu, Jun Wei, Lixin Wang, Alcohol dehydrogenase modulates quorum sensing in biofilm formations of Acinetobacter baumannii, 2020, 148, 08824010, 104451, 10.1016/j.micpath.2020.104451
    6. Anthony Adegoke, Adekunle Faleye, Gulshan Singh, Thor Stenström, Antibiotic Resistant Superbugs: Assessment of the Interrelationship of Occurrence in Clinical Settings and Environmental Niches, 2016, 22, 1420-3049, 29, 10.3390/molecules22010029
    7. Vijay Kothari, Chinmayi Joshi, Pooja Patel, Deepa Shahi, Charmi Mehta, Bhumika Prajapati, Sweta Patel, Dipeeka Mandaliya, Sriram Seshadri, 2018, 9780128136676, 325, 10.1016/B978-0-12-813667-6.00008-5
    8. K. Saipriya, C.H. Swathi, K.S. Ratnakar, V. Sritharan, Quorum‐sensing system in Acinetobacter baumannii : a potential target for new drug development , 2020, 128, 1364-5072, 15, 10.1111/jam.14330
    9. Yoshinori Sato, Yuka Unno, Tsuneyuki Ubagai, Yasuo Ono, Hendrik W. van Veen, Sub-minimum inhibitory concentrations of colistin and polymyxin B promote Acinetobacter baumannii biofilm formation, 2018, 13, 1932-6203, e0194556, 10.1371/journal.pone.0194556
    10. Fatemeh Hemmati, Roya Salehi, Reza Ghotaslou, Hossein Samadi Kafil, Alka Hasani, Pourya Gholizadeh, Roghayeh Nouri, Mohammad Ahangarzadeh Rezaee,

    Quorum Quenching: A Potential Target for Antipseudomonal Therapy

    , 2020, Volume 13, 1178-6973, 2989, 10.2147/IDR.S263196
    11. Niharika G. Jha, Daphika S. Dkhar, Sumit K. Singh, Shweta J. Malode, Nagaraj P. Shetti, Pranjal Chandra, Engineered Biosensors for Diagnosing Multidrug Resistance in Microbial and Malignant Cells, 2023, 13, 2079-6374, 235, 10.3390/bios13020235
    12. Evgenia Maslova, Lara Eisaiankhongi, Folke Sjöberg, Ronan R. McCarthy, Burns and biofilms: priority pathogens and in vivo models, 2021, 7, 2055-5008, 10.1038/s41522-021-00243-2
    13. Sowndarya Jothipandiyan, Devarajan Suresh, Sankaran Venkatachalam Sankaran, Subbiah Thamotharan, Kumaravel Shanmugasundaram, Preethi Vincent, Saravanan Sekaran, Shanmugaraj Gowrishankar, Shunmugiah Karutha Pandian, Nithyanand Paramasivam, Heteroleptic pincer palladium(II) complex coated orthopedic implants impede the AbaI/AbaR quorum sensing system and biofilm development by Acinetobacter baumannii, 2022, 38, 0892-7014, 55, 10.1080/08927014.2021.2015336
    14. Muhammad Hussnain Siddique, Sumreen Hayat, Saima Muzammil, Asma Ashraf, Arif Muhammad Khan, Muhammad Umar Ijaz, Mohsin Khurshid, Muhammad Afzal, Ecofriendly phytosynthesized zirconium oxide nanoparticles as antibiofilm and quorum quenching agents against Acinetobacter baumannii, 2022, 48, 0363-9045, 502, 10.1080/03639045.2022.2132260
    15. Sirijan Santajit, Nitat Sookrung, Nitaya Indrawattana, Quorum Sensing in ESKAPE Bugs: A Target for Combating Antimicrobial Resistance and Bacterial Virulence, 2022, 11, 2079-7737, 1466, 10.3390/biology11101466
    16. Veronica Lazar, Alina Maria Holban, Carmen Curutiu, Mariana Carmen Chifiriuc, Modulation of Quorum Sensing and Biofilms in Less Investigated Gram-Negative ESKAPE Pathogens, 2021, 12, 1664-302X, 10.3389/fmicb.2021.676510
    17. Rajat Kumar Jha, Rameez Jabeer Khan, Ekampreet Singh, Ankit Kumar, Monika Jain, Jayaraman Muthukumaran, Amit Kumar Singh, An extensive computational study to identify potential inhibitors of Acyl-homoserine-lactone synthase from Acinetobacter baumannii (strain AYE), 2022, 114, 10933263, 108168, 10.1016/j.jmgm.2022.108168
    18. Rajat Kumar Jha, Ekampreet Singh, Rameez Jabeer Khan, Ankit Kumar, Monika Jain, Jayaraman Muthukumaran, Amit Kumar Singh, Droperidol as a potential inhibitor of acyl-homoserine lactone synthase from A. baumannii: insights from virtual screening, MD simulations and MM/PBSA calculations, 2022, 1381-1991, 10.1007/s11030-022-10533-2
    19. Gustavo Mockaitis, Guillaume Bruant, Eugenio Foresti, Marcelo Zaiat, Serge R. Guiot, Physicochemical pretreatment selects microbial communities to produce alcohols through metabolism of volatile fatty acids, 2022, 2190-6815, 10.1007/s13399-022-02383-7
    20. Sagar Kiran Khadke, Jin-Hyung Lee, Yong-Guy Kim, Vinit Raj, Jintae Lee, Assessment of Antibiofilm Potencies of Nervonic and Oleic Acid against Acinetobacter baumannii Using In Vitro and Computational Approaches, 2021, 9, 2227-9059, 1133, 10.3390/biomedicines9091133
    21. Jwan Khidhr Rahman, Akhter Ahmed Ahmed, Aryan R. Ganjo, Trefa Salih Mohamad, Assessment of toxicity, anti-quorum sensing and anti-biofilm production effects of Hypericum triquetrifolium Turra extract on multi-drug resistant Acinetobacter baumannii, 2023, 35, 10183647, 102714, 10.1016/j.jksus.2023.102714
    22. Lizhi Li, Weikun Guan, Yihao Fan, Qin He, Dongsheng Guo, An Yuan, Qingfeng Xing, Yang Wang, Ziqin Ma, Jian Ni, Jia Chen, Qilong Zhou, Yuhong Zhong, Jiating Li, Haibo Zhang, Zinc/carbon nanomaterials inhibit antibiotic resistance genes by affecting quorum sensing and microbial community in cattle manure production, 2023, 387, 09608524, 129648, 10.1016/j.biortech.2023.129648
    23. Seetha Lakshmi Rajangam, Manoj Kumar Narasimhan, Current treatment strategies for targeting virulence factors and biofilm formation in Acinetobacter baumannii , 2024, 19, 1746-0913, 941, 10.2217/fmb-2023-0263
    24. Christiana E. Aruwa, Theolyn Chellan, Nosipho W. S'thebe, Yamkela Dweba, Saheed Sabiu, ESKAPE pathogens and associated quorum sensing systems: New targets for novel antimicrobials development, 2024, 11, 27726320, 100155, 10.1016/j.hsr.2024.100155
    25. Gabriel Gbenga Babaniyi, Babafemi Raphael Babaniyi, Ulelu Jessica Akor, 2023, 978-1-83916-761-4, 59, 10.1039/BK9781837671380-00059
    26. Andrea Marino, Egle Augello, Stefano Stracquadanio, Carlo Maria Bellanca, Federica Cosentino, Serena Spampinato, Giuseppina Cantarella, Renato Bernardini, Stefania Stefani, Bruno Cacopardo, Giuseppe Nunnari, Unveiling the Secrets of Acinetobacter baumannii: Resistance, Current Treatments, and Future Innovations, 2024, 25, 1422-0067, 6814, 10.3390/ijms25136814
    27. Sérgio G. Mendes, Sofia I. Combo, Thibault Allain, Sara Domingues, Andre G. Buret, Gabriela J. Da Silva, Co-regulation of biofilm formation and antimicrobial resistance in Acinetobacter baumannii: from mechanisms to therapeutic strategies, 2023, 42, 0934-9723, 1405, 10.1007/s10096-023-04677-8
    28. Mohammed Mansour Quradha, Alfred Ngenge Tamfu, Mehmet Emin Duru, Selcuk Kucukaydin, Mudassar Iqbal, Abdulkader Moqbel Farhan Qahtan, Rasool Khan, Ozgur Ceylan, Evaluation of HPLC Profile, Antioxidant, Quorum Sensing, Biofilm, Swarming Motility, and Enzyme Inhibition Activities of Conventional and Green Extracts of Salvia triloba, 2024, 2048-7177, 10.1002/fsn3.4580
    29. Abirami Karthikeyan, Manoj Kumar Thirugnanasambantham, Fazlurrahman Khan, Arun Kumar Mani, Bacteria-Inspired Synthesis of Silver-Doped Zinc Oxide Nanocomposites: A Novel Synergistic Approach in Controlling Biofilm and Quorum-Sensing-Regulated Virulence Factors in Pseudomonas aeruginosa, 2025, 14, 2079-6382, 59, 10.3390/antibiotics14010059
    30. Thejaswi Bhat, Manish Kumar, Krishna Kumar Ballamoole, Vijaya Kumar Deekshit, Pavan Gollapalli, Pangenome-based network analysis of Acinetobacter baumannii reveals the landscape of conserved therapeutic targets, 2025, 1381-1991, 10.1007/s11030-025-11252-0
  • Reader Comments
  • © 2015 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(10700) PDF downloads(1242) Cited by(1)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog