Review Special Issues

Neuroimmunomodulation by estrogen in health and disease

  • Systemic homeostasis is maintained by the robust bidirectional regulation of the neuroendocrine-immune network by the active involvement of neural, endocrine and immune mediators. Throughout female reproductive life, gonadal hormones undergo cyclic variations and mediate concomitant modulations of the neuroendocrine-immune network. Dysregulation of the neuroendocrine-immune network occurs during aging as a cumulative effect of declining neural, endocrine and immune functions and loss of compensatory mechanisms including antioxidant enzymes, growth factors and co-factors. This leads to disruption of homeostasis and sets the stage for the development of female-specific age-associated diseases such as autoimmunity, osteoporosis, cardiovascular diseases and hormone-dependent cancers. Ovarian hormones especially estrogen, play a key role in the maintenance of health and homeostasis by modulating the nervous, endocrine and immune functions and thereby altering neuroendocrine-immune homeostasis. Immunologically estrogen's role in the modulation of Th1/Th2 immune functions and contributing to pro-inflammatory conditions and autoimmunity has been widely studied. Centrally, hypothalamic and pituitary hormones influence gonadal hormone secretion in murine models during onset of estrous cycles and are implicated in reproductive aging-associated acyclicity. Loss of estrogen affects neuronal plasticity and the ensuing decline in cognitive functions during reproductive aging in females implicates estrogen in the incidence and progression of neurodegenerative diseases. Peripherally, sympathetic noradrenergic (NA) innervations of lymphoid organs and the presence of both adrenergic (AR) and estrogen receptors (ER) on lymphocytes poise estrogen as a potent neuroimmunomodulator during health and disease. Cyclic variations in estrogen levels throughout reproductive life, perimenopausal surge in estrogen levels followed by its precipitous decline, concomitant with decline in central hypothalamic catecholaminergic activity, peripheral sympathetic NA innervation and associated immunosuppression present an interesting study to explore female-specific age-associated diseases in a new light.

    Citation: Hannah P Priyanka, Rahul S Nair. Neuroimmunomodulation by estrogen in health and disease[J]. AIMS Neuroscience, 2020, 7(4): 401-417. doi: 10.3934/Neuroscience.2020025

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  • Systemic homeostasis is maintained by the robust bidirectional regulation of the neuroendocrine-immune network by the active involvement of neural, endocrine and immune mediators. Throughout female reproductive life, gonadal hormones undergo cyclic variations and mediate concomitant modulations of the neuroendocrine-immune network. Dysregulation of the neuroendocrine-immune network occurs during aging as a cumulative effect of declining neural, endocrine and immune functions and loss of compensatory mechanisms including antioxidant enzymes, growth factors and co-factors. This leads to disruption of homeostasis and sets the stage for the development of female-specific age-associated diseases such as autoimmunity, osteoporosis, cardiovascular diseases and hormone-dependent cancers. Ovarian hormones especially estrogen, play a key role in the maintenance of health and homeostasis by modulating the nervous, endocrine and immune functions and thereby altering neuroendocrine-immune homeostasis. Immunologically estrogen's role in the modulation of Th1/Th2 immune functions and contributing to pro-inflammatory conditions and autoimmunity has been widely studied. Centrally, hypothalamic and pituitary hormones influence gonadal hormone secretion in murine models during onset of estrous cycles and are implicated in reproductive aging-associated acyclicity. Loss of estrogen affects neuronal plasticity and the ensuing decline in cognitive functions during reproductive aging in females implicates estrogen in the incidence and progression of neurodegenerative diseases. Peripherally, sympathetic noradrenergic (NA) innervations of lymphoid organs and the presence of both adrenergic (AR) and estrogen receptors (ER) on lymphocytes poise estrogen as a potent neuroimmunomodulator during health and disease. Cyclic variations in estrogen levels throughout reproductive life, perimenopausal surge in estrogen levels followed by its precipitous decline, concomitant with decline in central hypothalamic catecholaminergic activity, peripheral sympathetic NA innervation and associated immunosuppression present an interesting study to explore female-specific age-associated diseases in a new light.


    Dedicated to Giuseppe Mingione on the occasion of his 50th birthday, with admiration.

    In this paper we provide a limiting partial regularity criterion for vector-valued minimizers u:ΩRnRN, n2, N>1, of nonhomogeneous, quasiconvex variational integrals as:

    W1,p(Ω;RN)wF(w;Ω):=Ω[F(Dw)fw]dx, (1.1)

    with standard p-growth. More precisely, we infer the optimal [31, Section 9] ε-regularity condition

    supBϱΩϱmBϱ|f|mdxε Du has a.e. bounded mean oscillation,

    and the related borderline function space criterion

    fL(n,)  supBϱΩϱmBϱ|f|mdxε.

    This is the content of our main theorem.

    Theorem 1.1. Under assumptions (1.6)1,2,3, (1.7) and (1.10), let uW1,p(Ω,RN) be a local minimizer of functional (1.1). Then, there exists a number εε(data)>0 such that if

    fLn,(Ω)(|B1|4n/m)1/nε, (1.2)

    then there exists an open set ΩuΩ with |ΩΩu|=0 such that

    DuBMOloc(Ωu;RN×n). (1.3)

    Moreover, the set Ωu can be characterized as follows

    Ωu:={x0Ω:εx0,ϱx0>0such that E(u;Bϱ(x0))εx0 for some ϱϱx0},

    where E() is the usual excess functional defined as

    E(w,z0;Bϱ(x0)):=( Bϱ(x0)|z0|p2|Dwz0|2+|Dwz0|p dx)1p. (1.4)

    We immediately refer to Section 1.2 below for a description of the structural assumptions in force in Theorem 1.1. Let us put our result in the context of the available literature. The notion of quasiconvexity was introduced by Morrey [38] in relation to the delicate issue of semicontinuity of multiple integrals in Sobolev spaces: an integrand F() is a quasiconvex whenever

    B1(0)F(z+Dφ)dxF(z)holds for all  zRN×n,  φCc(B1(0),RN). (1.5)

    Under power growth conditions, (1.5) is proven to be necessary and sufficient for the sequential weak lower semicontinuity on W1,p(Ω;RN); see [1,4,35,36,38]. It is worth stressing that quasiconvexity is a strict generalization of convexity: the two concepts coincide in the scalar setting (N=1), or for 1-d problems (n=1), but sharply differ in the multidimensional case: every convex function is quasiconvex thanks to Jensen's inequality, while the determinant is quasiconvex (actually polyconvex), but not convex, cf. [24, Section 5.1]. Another distinctive trait is the nonlocal nature of quasiconvexity: Morrey [38] conjectured that there is no condition involving only F() and a finite number of its derivatives that is both necessary and sufficient for quasiconvexity, fact later on confirmed by Kristensen [29]. A peculiarity of quasiconvex functionals is that minima and critical points (i.e., solutions to the associated Euler-Lagrange system) might have very different behavior under the (partial) regularity viewpoint. In fact, a classical result of Evans [22] states that the gradient of minima is locally Hölder continuous outside a negligible, " singular" set, while a celebrated counterexample due to Müller and Šverák [39] shows that the gradient of critical points may be everywhere discontinuous. After Evans seminal contribution [22], the partial regularity theory was extended by Acerbi and Fusco [2] to possibly degenerate quasiconvex functionals with superquadratic growth, and by Carozza, Fusco and Mingione [8] to subquadratic, nonsingular variational integrals. A unified approach that allows simultaneously handling degenerate/nondegenerate, and singular/nonsingular problems, based on the combination of A-harmonic approximation [21], and p-harmonic approximation [20], was eventually proposed by Duzaar and Mingione [19]. Moreover, Kristensen and Mingione [30] proved that the Hausdorff dimension of the singular set of Lipschitz continuous minimizers of quasiconvex multiple integrals is strictly less than the ambient space dimension n, see also [5] for further developments in this direction. We refer to [3,15,16,25,26,27,28,37,41,42] for an (incomplete) account of classical, and more recent advances in the field. In all the aforementioned papers are considered homogeneous functionals, i.e., f0 in (1.1). The first sharp ε-regularity criteria for nonhomogeneous quasiconvex variational integrals guaranteeing almost everywhere gradient continuity under optimal assumptions on f were obtained by De Filippis [12], and De Filippis and Stroffolini [14], by connecting the classical partial regularity theory for quasiconvex functionals with nonlinear potential theory for degenerate/singular elliptic equations, first applied in the context of partial regularity for strongly elliptic systems by Kuusi and Mingione [33]. Potential theory for nonlinear PDE originates from the classical problem of determining the best condition on f implying gradient continuity in the Poisson equation Δu=f, that turns out to be formulated in terms of the uniform decay to zero of the Riesz potential, in turn implied by the membership of f to the Lorentz space L(n,1), [9,31]. In this respect, a breakthrough result due to Kuusi and Mingione [32,34] states that the same is true for the nonhomogeous, degenerate p-Laplace equation–in other words, the regularity theory for the nonhomogeneous p-Laplace PDE coincides with that of the Poisson equation up to the C1-level. This important result also holds in the case of singular equations [18,40], for general, uniformly elliptic equations [6], up to the boundary [10,11,13], and at the level of partial regularity for p-Laplacian type systems without Uhlenbeck structure, [7,33]. We conclude by highlighting that our Theorem 1.1 fits this line of research as, it determines for the first time in the literature optimal conditions on the inhomogeneity f assuring partial BMO-regularity for minima of quasiconvex functionals expressed in terms of the limiting function space L(n,).

    In Section 2 we recall some well-known results from the study of nonlinear problems also establishing some Caccioppoli and Gehring type lemmas. In Section 3 we prove the excess decay estimates; considering separately the nondegenerate and the degenerate case. Section 4 is devoted to the proof of Theorem 1.1.

    In (1.1), the integrand F:RN×nR satisfies

    { FC2loc(RN×n) Λ1|z|pF(z)Λ|z|p |2F(z)|Λ|z|p2 |2F(z1)2F(z2)|μ(|z2z1||z2|+|z1|)(|z1|2+|z2|2)p22 (1.6)

    for all zRN×n, Λ1 being a positive absolute constant and μ:[0,)[0,1] being a concave nondecreasing function with μ(0)=0. In the rest of the paper we will always assume p2. In order to derive meaningful regularity results, we need to update (1.5) to the stronger strict quasiconvexity condition

    B[F(z+Dφ)F(z)] dxλB(|z|2+|Dφ|2)p22|Dφ|2dx, (1.7)

    holding for all zRN×n and φW1,p0(B,RN), with λ being a positive, absolute constant. Furthermore, we allow the integrand F() to be degenerate elliptic in the origin. More specifically, we assume that F() features degeneracy of p-Laplacian type at the origin, i.e.,

    | F(z)F(0)|z|p2z|z|p1 |0as|z|0, (1.8)

    which means that we can find a function ω:(0,)(0,) such that

    |z|ω(s)  |F(z)F(0)|z|p2z|s|z|p1, (1.9)

    for every zRN×n and all s(0,). Moreover, the right-hand side term f:ΩRN in (1.1) verifies as minimal integrability condition the following

    fLm(Ω,RN)with  2>m>{ 2n/(n+2)if  n>2, 3/2if  n=2, (1.10)

    which, being p2, in turn implies that

    fW1,p(Ω,RN)andm<2p. (1.11)

    Here it is intended that, when pn, the Sobolev conjugate exponent p can be chosen as large as needed - in particular it will always be larger than p. By (1.5) and (1.6)2 we have

    |F(z)|c|z|p1, (1.12)

    with cc(n,N,Λ,p); see for example [35, proof of Theorem 2.1]. Finally, (1.7) yields that for all zRN×n, ξRN, ζRn it is

    2F(z)ξζ,ξζ2λ|z|p2|ξ|2|ζ|2, (1.13)

    see [24, Chapter 5].

    In this section we display our notation and collect some basic results that will be helpful later on.

    In this paper, ΩRn is an open, bounded domain with Lipschitz boundary, and n2. By c we will always denote a general constant larger than one, possibly depending on the data of the problem. Special occurrences will be denoted by c,˜c or likewise. Noteworthy dependencies on parameters will be highlighted by putting them in parentheses. Moreover, to simplify the notation, we shall array the main parameters governing functional (1.1) in the shorthand data:=(n,N,λ,Λ,p,μ(),ω()). By Br(x0):={xRn:|xx0|<r}, we denote the open ball with radius r, centred at x0; when not necessary or clear from the context, we shall omit denoting the center, i.e., Br(x0)Br - this will happen, for instance, when dealing with concentric balls. For x0Ω, we abbreviate dx0:=min{1,dist(x0,Ω)}. Moreover, with BRn being a measurable set with bounded positive Lebesgue measure 0<|B|<, and a:BRk, k1, being a measurable map, we denote

    (a)BBa(x) dx:=1|B|Ba(x)dx.

    We will often employ the almost minimality property of the average, i.e.,

    ( B|a(a)B|tdx)1/t2( B|az|tdx)1/t (2.1)

    for all zRN×n and any t1. Finally, if t>1 we will indicate its conjugate by t:=t/(t1) and its Sobolev exponents as t:=nt/(nt) if t<n or any number larger than one for tn and t:=max{nt/(n+t),1}.

    When dealing with p-Laplacian type problems, we shall often use the auxiliary vector field Vs:RN×nRN×n, defined by

    Vs(z):=(s2+|z|2)(p2)/4zwithp(1,),  s0,  zRN×n,

    incorporating the scaling features of the p-Laplacian. If s=0 we simply write Vs()V(). A couple of useful related inequalities are

    { |Vs(z1)Vs(z2)|(s2+|z1|2+|z2|2)(p2)/4|z1z2|, |Vs(z1+z2)||Vs(z1)|+|Vs(z2)|, |Vs1(z)||Vs2(z)|, if 12s2s12s2, |V(z1)V(z2)|2|V|z1|(z1z2)|2, if 12|z2||z1|2|z2|, (2.2)

    and

    |Vs(z)|2sp2|z|2+|z|pwith  p2, (2.3)

    where the constants implicit in " ", " " depend on n,N,p. A relevant property which is relevant for the nonlinear setting is recorded in the following lemma.

    Lemma 2.1. Let t>1, s[0,1] and z1,z2RN×n be such that s+|z1|+|z2|>0. Then

    10[s2+|z1+y(z2z1)|2]t2 dy(s2+|z1|2+|z2|2)t2,

    with constants implicit in "" depending only on n,N,t.

    The following iteration lemma will be helpful throughout the rest of the paper; for a proof we refer the reader to [24, Lemma 6.1].

    Lemma 2.2. Let h:[ϱ0,ϱ1]R be a non-negative and bounded function, and let θ(0,1), A,B,γ1,γ20 be numbers. Assume that h(t)θh(s)+A(st)γ1+B(st)γ2 holds for all ϱ0t<sϱ1. Then the following inequality holds h(ϱ0)c(θ,γ1,γ2)[A(ϱ1ϱ0)γ1+B(ϱ1ϱ0)γ2].

    We will often consider the "quadratic" version of the excess functional defined in (1.4), i.e.,

    ˜E(w,z0;Bϱ(x0)):=( Bϱ(x0)|V(Dw)z0|2dx)12. (2.4)

    In the particular case z0=(Dw)Bϱ(x0) (z0=(V(Dw))Bϱ(x0), resp.) we shall simply write E(w,(Dw)Bϱ(x0);Bϱ(x0))E(w;Bϱ(x0)) (˜E(w,(V(Dw))Bϱ(x0);Bϱ(x0))˜E(w;Bϱ(x0)), resp.). A simple computation shows that

    E(w;Bϱ(x0))p/2˜E(w;Bϱ(x0)). (2.5)

    Moreover, from (2.1) and from [23, Formula (2.6)] we have that

    ˜E(w;Bϱ(x0))˜E(w,V((Dw)Bϱ(x0));Bϱ(x0)). (2.6)

    In this section we collect some basic estimates for local minimizers of nonhomogeneous quasiconvex functionals. We start with a variation of the classical Caccioppoli inequality accounting for the presence of a nontrivial right-hand side term, coupled with an higher integrability result of Gehring-type.

    Lemma 2.3. Under assumptions (1.6)1,2,3, (1.7) and (1.10), let uW1,p(Ω,RN) be a local minimizer of functional (1.1).

    For every ball Bϱ(x0)Ω and any u0RN, z0RN×n{0} it holds that

    E(u,z0;Bϱ/2(x0))pcBϱ(x0)|z0|p2|uϱ|2+|uϱ|pdx+c|z0|p2(ϱmBϱ(x0)|f|m dx)2m, (2.7)

    where E() is defined in (1.4), (x):=u0+z0,xx0 and cc(n,N,λ,Λ,p).

    There exists an higher integrability exponent p2p2(n,N,λ,Λ,p)>p such that DuLp2loc(Ω,RN×n) and the reverse Hölder inequality

    ( Bϱ/2(x0)|Du(Du)Bϱ(x0)|p2dx)1p2c( Bϱ(x0)|Du|pdx)1p+c(ϱmBϱ(x0)|f|mdx)1m(p1), (2.8)

    is verified for all balls Bϱ(x0)Ω with cc(n,N,λ,Λ,p).

    Proof. For the ease of exposition, we split the proof in two steps, each of them corresponding to the proof of (2.7) and (2.8) respectively.

    Step 1: proof of (2.7).

    We choose parameters ϱ/2τ1<τ2ϱ, a cut-off function ηC1c(Bτ2(x0)) such that 1Bτ1(x0)η1Bτ2(x0) and |Dη|(τ2τ1)1. Set φ1:=η(u), φ2:=(1η)(u) and use (1.7) and the equivalence in (2.2)1 to estimate

    cBτ2(x0)|V|z0|(Dφ1)|2 dxBτ2(x0)[F(z0+Dφ1)F(z0)]dx=Bτ2(x0)[F(DuDφ2)F(Du)]dx+Bτ2(x0)[F(Du)F(DuDφ1)]dx+Bτ2(x0)[F(z0+Dφ2)F(z0)]dx=:I1+I2+I3, (2.9)

    where we have used the simple relation Dφ1+Dφ2=Duz0. Terms I1 and I3 can be controlled as done in [19, Proposition 2]; indeed we have

    I1+I3cBτ2(x0)Bτ1(x0)|V|z0|(Dφ2)|2dx+cBτ2(x0)Bτ1(x0)|V|z0|(Duz0)|2dx(2.2)2cBτ2(x0)Bτ1(x0)|V|z0|(Duz0)|2+| V|z0|(uτ2τ1) |2dx, (2.10)

    for cc(n,N,λ,Λ,p). Concerning term I2, we exploit (1.10), the fact that φ1W1,p0(Bτ2(x0),RN) and apply Sobolev-Poincaré inequality to get

    I2|Bτ2(x0)|(τm2Bτ2(x0)|f|m dx)1/m(τm2Bτ2(x0)|φ1|m dx)1m|Bτ2(x0)|(τm2Bτ2(x0)|f|m dx)1/m( Bτ2(x0)| φ1τ2 |2 dx)12|Bτ2(x0)|(τm2Bτ2(x0)|f|mdx)1/m( Bτ2(x0)|Dφ1|2dx)12εBτ2(x0)|V|z0|(Dφ1)|2dx+c|Bϱ(x0)|ε|z0|p2(ϱmBϱ(x0)|f|mdx)2m, (2.11)

    where cc(n,N,m) and we also used that ϱ/2τ2ϱ. Merging the content of the two above displays, recalling that η1 on Bτ1(x0) and choosing ε>0 sufficiently small, we obtain

    Bτ1(x0)|V|z0|(Duz0)|2dxcBτ2(x0)Bτ1(x0)|V|z0|(Duz0)|2+| V|z0|(uτ2τ1) |2dx+c|Bϱ(x0)||z0|p2(ϱmBϱ(x0)|f|mdx)2m,

    with cc(n,N,λ,Λ,p). At this stage, the classical hole-filling technique, Lemma 2.2 and (2.3) yield (2.7) and the first bound in the statement is proven.

    Step 2: proof of (2.8).

    To show the validity of (2.8), we follow [33, proof of Proposition 3.2] and first observe that if u is a local minimizer of functional F() on Bϱ(x0), setting fϱ(x):=ϱf(x0+ϱx), the map uϱ(x):=ϱ1u(x0+ϱx) is a local minimizer on B1(0) of an integral with the same integrand appearing in (1.1) satisfying (1.6)1,2,3 and fϱ replacing f. This means that (2.10) still holds for all balls Bσ/2(˜x)Bτ1(˜x)Bτ2(˜x)Bσ(˜x)B1(0), with ˜xB1(0) being any point, in particular it remains true if |z0|=0, while condition |z0|0 was needed only in the estimate of term I2 in (2.11), that now requires some change. So, in the definition of the affine map we choose z0=0, u0=(uϱ)Bσ(˜x) and rearrange estimates (2.10) and (2.11) as:

    I1+I3(2.3)cBτ2(˜x)Bτ1(˜x)|Duϱ|p+| uϱ(uϱ)Bσ(˜x)τ2τ1 |pdx,

    and, recalling that φ1W1,p0(Bτ2(˜x),RN), via Sobolev Poincaré, Hölder and Young inequalities and (1.11)2, we estimate

    I2|Bτ2(˜x)|(τ(p)2Bτ2(˜x)|fϱ|(p) dx)1(p)(τp2Bτ2(˜x)|φ1|p dx)1pc|Bτ2(˜x)|(τ(p)2Bτ2(˜x)|fϱ|(p)dx)1(p)( Bτ2(˜x)|Dφ1|pdx)1pc|Bσ(˜x)|ε1/(p1)(σ(p)Bσ(˜x)|fϱ|(p)dx)p(p)(p1)+εBτ2(˜x)|Dφ1|pdx,

    with cc(n,N,p). Plugging the content of the two previous displays in (2.9), reabsorbing terms and applying Lemma 2.2, we obtain

    Bσ/2(˜x)|Duϱ|p dxc Bσ(˜x)| uϱ(uϱ)Bσ(˜x)σ |p dx+c(σ(p)Bσ(˜x)|fϱ|(p)dx)p(p)(p1), (2.12)

    for cc(n,N,Λ,λ,p). Notice that

    n(p(p)(p1)1)pp1, (2.13)

    with equality holding when p<n, while for pn any value of p>1 will do. We then manipulate the second term on the right-hand side of (2.12) as

    (σ(p)Bσ(˜x)|fϱ|(p)dx)p(p)(p1)σpp1n(p(p)(p1)1)( B1(0)|fϱ|(p) dx)p(p)(p1)1Bσ(˜x)|fϱ|(p)dx(2.13)( B1(0)|fϱ|(p) dx)p(p)(p1)1Bσ(˜x)|fϱ|(p) dx=:Bσ(˜x)|Kϱfϱ|(p) dx,

    where we set

    K(p)ϱ:=|B1(0)|1p(p)(p1)fϱpp1(p)L(p)(B1(0)).

    Plugging the content of the previous display in (2.12) and applying Sobolev-Poincaré inequality we get

    Bσ/2(˜x)|Duϱ|pdxc( Bσ(˜x)|Duϱ|p dx)pp+c Bσ(˜x)|Kϱfϱ|(p) dx,

    with cc(n,N,Λ,λ,p). Now we can apply a variant of Gehring lemma [24, Corollary 6.1] to determine a higher integrability exponent ss(n,N,Λ,λ,p) such that 1<sm/(p) and

    ( Bσ/2(˜x)|Duϱ|spdx)1spc( Bσ(˜x)|Duϱ|pdx)1p+cK(p)/pϱ( Bσ(˜x)|fϱ|s(p) dx)1sp

    for cc(n,N,Λ,λ,p). Next, notice that

    K(p)/pϱ=( B1(0)|fϱ|(p)dx)1(p)(p1)1p( B1(0)|fϱ|s(p)dx)1s(p)(p1)1sp,

    so plugging this last inequality in (2.14) and recalling that s(p)m, we obtain

    ( Bσ/2(˜x)|Duϱ|spdx)1spc( Bσ(˜x)|Duϱ|pdx)1p+c( Bσ(˜x)|fϱ|m dx)1m(p1).

    Setting p2:=sp>p above and recalling that ˜xB1(0) is arbitrary, we can fix ˜x=0, scale back to Bϱ(x0) and apply (2.1) to get (2.8) and the proof is complete.

    In this section we prove some excess decay estimates considering separately two cases: when a smallness condition on the excess functional of our local minimizer u is satisfied and when such an estimate does not hold true.

    We start working assuming that a suitable smallness condition on the excess functional E(u;Bϱ(x0)) is fulfilled. In particular, we prove the following proposition.

    Proposition 3.1. Under assumptions (1.6)1,2,3, (1.7) and (1.10), let uW1,p(Ω,RN) be a local minimizer of functional (1.1). Then, for τ0(0,210), there exists ε0ε0(data,τ0)(0,1) and ε1ε1(data,τ0)(0,1) such that the following implications hold true.

    If the conditions

    E(u;Bϱ(x0))ε0|(Du)Bϱ(x0)|, (3.1)

    and

    (ϱmBϱ(x0)|f|m dx)1mε1|(Du)Bϱ(x0)|p22E(u;Bϱ(x0))p2, (3.2)

    are verified on Bϱ(x0), then it holds that

    E(u;Bτ0ϱ(x0))c0τβ00E(u;Bϱ(x0)), (3.3)

    for all β0(0,2/p), with c0c0(data)>0.

    If condition (3.1) holds true and

    (ϱmBϱ(x0)|f|m dx)1m>ε1|(Du)Bϱ(x0)|p22E(u;Bϱ(x0))p2, (3.4)

    is satisfied on Bϱ(x0), then

    E(u;Bτ0ϱ(x0))c0(ϱmBϱ(x0)|f|m dx)1m(p1), (3.5)

    for c0c0(data)>0.

    Proof of Proposition 3.1. For the sake of readability, since all balls considered here are concentric to Bϱ(x0), we will omit denoting the center. Moreover, we will adopt the following notation (Du)Bς(x0)(Du)ς and, for all φCc(Bϱ;RN), we will denote DφL(Bϱ)Dφ. We spilt the proof in two steps.

    Step 1: proof of (3.3).

    With no loss of generality we can assume that E(u;Bϱ)>0, which clearly implies, thanks to (3.1), that |(Du)ϱ|>0.

    We begin proving that condition (3.1) implies that

    Bϱ|Du|pdxc|(Du)ϱ|p, (3.6)

    for a constant cc(p,ε0)>0. Indeed,

    Bϱ|Du|pdxc Bϱ|Du(Du)ϱ|pdx+c|(Du)ϱ|p(1.4)cE(u;Bϱ)p+c|(Du)ϱ|p(3.1)c(εp0+1)|(Du)ϱ|p,

    and (3.6) follows.

    Consider now

    Bϱxu0(x):=|(Du)ϱ|p22(u(x)(u)ϱ(Du)ϱ,xx0)E(u;Bϱ)p/2, (3.7)

    and

    d:=(E(u;Bϱ)|(Du)ϱ|)p2.

    Let us note that we have

    Bϱ|Du0|2dx+dp2Bϱ|Du0|pdx|(Du)ϱ|p2E(u;Bϱ)pBϱ|Du(Du)ϱ|2dx+(E(u;Bϱ)|(Du)ϱ|)p(p2)2|(Du)ϱ|p(p2)2E(u;Bϱ)p22Bϱ|Du(Du)ϱ|pdx1E(u;Bϱ)pBϱ|(Du)ϱ|p2|Du(Du)ϱ|2dx+1E(u;Bϱ)pBϱ|Du(Du)ϱ|pdx1.

    Since |(Du)ϱ|>0 we have that the hypothesis of [12, Lemma 3.2] are satisfied with

    A:=2F((Du)ϱ)|(Du)ϱ|2p. (3.8)

    Then,

    | BϱADu0,Dφ dx|cDφ|(Du)ϱ|2p2E(u;Bϱ)p2(ϱmBϱ|f|m dx)1m+cDφμ(E(u;Bϱ)|(Du)ϱ|)1p[1+(E(u;Bϱ)|(Du)ϱ|)p22](3.1),(3.2)cε1Dφ+cDφμ(ε0)1p[1+εp220].

    Fix ε>0 and let δδ(data,ε)>0 be the one given by [33, Lemma 2.4] and choose ε0 and ε1 sufficiently small such that

    cε1+cμ(ε0)1p[1+εp220]δ. (3.9)

    With this choice of ε0 and ε1 it follows that u0 is almost A-harmonic on Bϱ, in the sense that

    | BϱADu0,Dφ dx|δDφ,

    with A as in (3.8). Hence, by [33, Lemma 2.4] we obtain that there exists h0W1,2(Bϱ;RN) which is A-harmonic, i.e.,

    BϱADh0,Dφ dx=0for all φCc(Bϱ;RN),

    such that

    B3ϱ/4|Dh0|2dx+dp2B3ϱ/4|Dh0|pdx82np, (3.10)

    and

    B3ϱ/4|u0h0ϱ|2+dp2|u0h0ϱ|pdxε. (3.11)

    We choose now τ0(0,210), which will be fixed later on, and estimate

    B2τ0ϱ|u0(x)h0(x0)Dh0(x0),xx0τ0ϱ|2dxcB2τ0ϱ|h0(x)h0(x0)Dh0(x0),xx0τ0ϱ|2dx+cB2τ0ϱ|u0h0τ0ϱ|2dx(3.11)c(τ0ϱ)2supBϱ/2|D2h0|2+cετn+20cτ20B3ϱ/4|Dh0|2dx+cετn+20(3.10)cτ20+cετn+20, (3.12)

    where cc(data)>0 and where we have used the following property of A-harmonic functions

    ϱγsupBϱ/2|D2h0|γcB3ϱ/4|Dh0|γdx, (3.13)

    with γ>1 and c depending on n, N, and on the ellipticity constants of A.

    Now, choosing

    ε:=τn+2p0,

    we have that this together with (3.9) gives that ε0ε0(data,τ0) and ε1ε1(data,τ0). Recalling the definition of u0 in (3.7) and (3.12) we eventually arrive at

    B2τ0ϱ|u(u)ϱ(Du)ϱ,xx0|(Du)ϱ|2p2E(u;Bϱ)p/2(h0(x0)Dh0(x0),xx0)|2(τ0ϱ)2dxc|(Du)ϱ|2pE(u;Bϱ)pτ20, (3.14)

    for cc(data)>0. By a similar computation, always using (3.13), (3.10) and (3.11), we obtain that

    dp2B2τ0ϱ|u0h0(x0)Dh0(x0),xx0τ0ϱ|pdxcdp2(τ0ϱ)psupBϱ/2|D2h0|p+cετn+p0cτp0.

    In this way, as for (3.14), by the definition of u0 in (3.7), we eventually arrive at

    B2τ0ϱ|u(u)ϱ(Du)ϱ,xx0|(Du)ϱ|2p2E(u;Bϱ)p/2(h0(x0)Dh0(x0),xx0)|p(τ0ϱ)pdxcd2p|(Du)ϱ|p(2p)2E(u;Bϱ)p22τp0cE(u;Bϱ)pτ20, (3.15)

    with cc(data).

    Denote now with 2τ0ϱ the unique affine function such that

    2τ0ϱmin affineB2τ0ϱ|u|2dx.

    Hence, by (3.14) and (3.15), we conclude that

    B2τ0ϱ|(Du)ϱ|p2|u2τ0ϱ2τ0ϱ|2+|u2τ0ϱ2τ0ϱ|pdxcτ2E(u;Bϱ)p. (3.16)

    Notice that we have also used the property that

    Bϱ|uϱ|pdxcBϱ|u|pdx,

    for p2, cc(n,N,p)>0 and for any affine function ; see [33, Lemma 2.3].

    Recalling the definition of the excess functional E(), in (1.4), we can estimate the following quantity as follows

    |D2τ0ϱ(Du)ϱ||D2τ0ϱ(Du)2τ0ϱ|+|(Du)2τ0ϱ(Du)ϱ|c(B2τ0ϱ|Du(Du)2τ0ϱ|2dx)12+(B2τ0ϱ|Du(Du)ϱ|2dx)12(2.1)cτn/20(Bϱ|Du(Du)ϱ|2dx)12=c|(Du)ϱ|2p2τn/20(Bϱ|(Du)ϱ|p2|Du(Du)ϱ|2dx)12c(n)τn/20(E(u,Bϱ)|(Du)ϱ|)p2|(Du)ϱ|, (3.17)

    where we have used the following property of the affine function 2τ0ϱ

    |D2τ0ϱ(Du)2τ0ϱ|pcB2τ0ϱ|Du(Du)2τ0ϱ|pdx,

    for a constant cc(n,p)>0; see for example [33, Lemma 2.2].

    Now, starting from (3.1) and (3.9), we further reduce the size of ε0 such that

    (E(u,Bϱ)|(Du)ϱ|)p2(3.1)εp20τn/208c(n), (3.18)

    where cc(n) is the same constant appearing in (3.17). Thus, combining (3.17) and (3.18), we get

    |D2τ0ϱ(Du)ϱ||(Du)ϱ|8. (3.19)

    The information provided by (3.18) combined with (3.16) allow us to conclude that

    B2τ0ϱ|D2τ0ϱ|p2|u2τ0ϱ2τ0ϱ|2+|u2τ0ϱ2τ0ϱ|pdxcτ2E(u;Bϱ)p. (3.20)

    By triangular inequality and (3.19) we also get

    |D2τ0ϱ||(Du)ϱ||D2τ0ϱ(Du)ϱ|(3.19)7|(Du)ϱ|8

    which, therefore, implies that

    Bτ0ϱ|D2τ0ϱ|p2|DuD2τ0ϱ|2dx+infzRN×nBτ0ϱ|Duz|pdx(2.7)cB2τ0ϱ|D2τ0ϱ|p2|u2τ0ϱ2τ0ϱ|2+|u2τ0ϱ2τ0ϱ|pdx+c|D2τ0ϱ|p2((2τ0ϱ)mB2τ0ϱ|f|mdx)2m(3.20)cτ20E(u,Bϱ)p+cτ22n/m0|(Du)ϱ|p2(ϱmBϱ|f|mdx)2m, (3.21)

    where cc(data)>0. By triangular inequality, we can further estimate

    Bτ0ϱ|(Du)τ0ϱ|p2|Du(Du)τ0ϱ|2dxcBτ0ϱ|Dτ0ϱ(Du)τ0ϱ|p2|Du(Du)τ0ϱ|2dx+cBτ0ϱ|D2τ0ϱDτ0ϱ|p2|Du(Du)τ0ϱ|2dx+cBτ0ϱ|D2τ0ϱ|p2|Du(Du)τ0ϱ|2dx=I1+I2+I3,

    where cc(p)>0. We now separately estimate the previous integrals. We begin considering I1. By Young and triangular inequalities we get

    I1c|Dτ0ϱ(Du)τ0ϱ|p+cBτ0ϱ|Du(Du)τ0ϱ|pdxcBτ0ϱ|Du(Du)τ0ϱ|pdx(2.1)cinfzRNBτ0ϱ|Duz|pdx(3.21)cτ20E(u,Bϱ)p+cτ22n/m0|(Du)ϱ|p2(ϱmBϱ|f|mdx)2m,

    with cc(data)>0. In a similar fashion, we can treat the integral I2

    I2c|D2τ0ϱDτ0ϱ|p+cBτ0ϱ|Du(Du)τ0ϱ|pdx(2.1)cB2τ0ϱ|u2τ0ϱ2τ0ϱ|pdx+cinfzRN×nBτ0ϱ|Duz|pdx(3.20),(3.21)cτ20E(u,Bϱ)p+cτ22n/m0|(Du)ϱ|p2(ϱmBϱ|f|mdx)2m,

    where we have used the following property of the affine function 2τ0ϱ

    |D2τ0ϱDτ0ϱ|pcB2τ0ϱ|u2τ0ϱ2τ0ϱ|pdx,

    for a given constant cc(n,p)>0; see [33, Lemma 2.2]. Finally, the last integral I3 can be treated recalling (3.21) and (2.1), i.e.,

    I3cτ20E(u,Bϱ)p+cτ22n/m0|(Du)ϱ|p2(ϱmBϱ|f|mdx)2m.

    All in all, combining the previous estimate

    E(u;Bτ0ϱ)cτ2/p0E(u,Bϱ)+cτ2/p2n/(mp)0|(Du)ϱ|p2p(ϱmBϱ|f|mdx)2mp(3.2)cτ2/p0E(u,Bϱ)+cτ2/p2n/(mp)0ε2/p1E(u;Bτ0ϱ)c0τ2/p0E(u;Bτ0ϱ),

    up to choosing ε1 such that

    ε1τn/m0.

    Step 2: proof of (3.5).

    The proof follows by [12, Lemma 2.4] which yields

    E(u;Bτ0ϱ(x0))p223pτn/20E(u;Bϱ(x0))p2(3.4)23pτn/20ε11|(Du)Bϱ(x0)|2p2(ϱmBϱ(x0)|f|m dx)1m(3.1)26(p1)τn(p1)/p0εp220ε11E(u;Bτ0ϱ)2p2(ϱmBϱ(x0)|f|m dx)1m.

    Multiplying both sides by E(u;Bτ0ϱ)p22 we get the desired estimate.

    It remains to considering the case when condition (3.1) does not hold true. We start with two technical lemmas. The first one is an analogous of the Caccioppoli inequality (2.7), where we take in consideration the eventuality z0=0.

    Lemma 3.1. Under assumptions (1.6)1,2,3, (1.7) and (1.10), let uW1,p(Ω,RN) be a local minimizer of functional (1.1). For every ball Bϱ(x0)Ω and any u0RN, z0RN×n it holds that

    E(u,z0;Bϱ/2(x0))pc Bϱ(x0)|z0|p2|uϱ|2+|uϱ|p dx+c(ϱmBϱ(x0)|f|m dx)pm(p1), (3.22)

    where E() is defined in (1.4), (x):=u0+z0,xx0 and cc(n,N,λ,Λ,p).

    Proof. The proof is analogous to estimate (2.7), up to treating in a different way the term I2 in (2.9), taking in consideration the eventuality z0=0. Exploiting (1.10) and fact that φ1W1,p0(Bτ2(x0),RN), an application of the Sobolev-Poincaré inequality yields

    I2|Bτ2(x0)|(τm2Bτ2(x0)|f|m dx)1/m(τm2Bτ2(x0)|φ1|m dx)1m|Bτ2(x0)|(τm2Bτ2(x0)|f|m dx)1/m( Bτ2(x0)| φ1τ2 |p dx)1p|Bτ2(x0)|(τm2Bτ2(x0)|f|mdx)1/m( Bτ2(x0)|Dφ1|pdx)1pεBτ2(x0)|V|z0|(Dφ1)|2dx+c|Bϱ(x0)|ε1/(p1)(ϱmBϱ(x0)|f|mdx)pm(p1), (3.23)

    where cc(n,N,m) and we also used that ϱ/2τ2ϱ. Hence, proceeding as in the proof of (2.7), we obtain that

    Bτ1(x0)|V|z0|(Duz0)|2dxcBτ2(x0)Bτ1(x0)|V|z0|(Duz0)|2+| V|z0|(uτ2τ1) |2dx+c|Bϱ(x0)|ε1/(p1)(ϱmBϱ(x0)|f|mdx)pm(p1),

    with cc(n,N,λ,Λ,p). Concluding as in the proof of (2.7), we eventually arrive at (3.22). We will also need the following result.

    Lemma 3.2. Under assumptions (1.6)1,2,3, (1.7) and (1.10), let uW1,p(Ω,RN) be a local minimizer of functional (1.1). For any Bϱ(x0)Ω and any s(0,) it holds that

    | Bϱ(x0)|Du|p2Du,Dφ dx|sDφL(Bϱ(x0))( Bϱ(x0)|Du|p dx)p1p+cω(s)1DφL(Bϱ(x0))Bϱ(x0)|Du|p dx+cDφL(Bϱ(x0))(ϱmBϱ(x0)|f|m dx)1/m, (3.24)

    for any φC0(Bϱ(x0),RN), with cc(n,N,Λ,λ,p).

    Proof. Given the regularity properties of the integrand F, we have that a local minimizer u of (1.1) solves weakly the following integral identity (see [42, Lemma 7.3])

    Ω[F(Du),Dφfφ] dx=0for all φC0(Ω,RN). (3.25)

    Now, fix φC0(Bϱ(x0),RN) and split

    | Bϱ(x0)|Du|p2Du,Dφ dx|(3.25)| Bϱ(x0)F(Du)F(0)|Du|p2Du,Dφ dx|+| Bϱ(x0)fφ dx|=:I1+I2.

    We begin estimating the first integral I1. For s(0,) we get

    I1DφL(Bϱ(x0))|Bϱ(x0)|Bϱ(x0){|Du|ω(s)}|F(Du)F(0)|Du|p2Du| dx+DφL(Bϱ(x0))|Bϱ(x0)|Bϱ(x0){|Du|>ω(s)}|F(Du)F(0)|Du|p2Du| dxsDφL(Bϱ(x0))( Bϱ(x0)|Du|p dx)p1p+cω(s)1DφL(Bϱ(x0))Bϱ(x0)|Du|p dx. (3.26)

    On the other hand, the integral I2 can be estimated as follows

    I2(ϱmBϱ(x0)|f|m dx)1/m( Bϱ(x0)|φϱ|m dx)1m(ϱmBϱ(x0)|f|m dx)1/m( Bϱ(x0)| φϱ |p dx)1p(ϱmBϱ(x0)|f|mdx)1/m( Bϱ(x0)|Dφ|pdx)1pDφL(Bϱ(x0))(ϱmBϱ(x0)|f|mdx)1/m.

    Combining the inequalities above we obtain (3.24).

    In this setting the analogous result of Proposition 3.1 is the following one.

    Proposition 3.2. Under assumptions (1.6)1,2,3, (1.7) and (1.10), let uW1,p(Ω,RN) be a local minimizer of functional (1.1). Then, for any χ(0,1] and any τ1(0,210), there exists ε2ε2(data,χ,τ1)(0,1) such that if the smallness conditions

    χ|(Du)Bϱ(x0)|E(u;Bϱ(x0)),andE(u;Bϱ(x0))ε2, (3.27)

    are satisfied on a ball Bϱ(x0)Rn, then

    E(u;Bτ1ϱ(x0))c1τβ11E(u;Bϱ(x0))+c1(ϱmBϱ(x0)|f|m dx)1m(p1), (3.28)

    for any β1(0,2α/p), with αα(n,N,p)(0,1) is the exponent in (3.34), and c1c1(data,χ).

    Proof. We adopt the same notations used in the proof of Proposition 3.1. Let us begin noticing that condition (3.27)1 implies the following estimate

    Bϱ|Du|p dxcχE(u;Bϱ)pwithcχ:=2p(1+χp). (3.29)

    Indeed, by (1.4) and (3.27), we have

    Bϱ|Du|p dx2pBϱ|Du(Du)Bϱ|p dx+2p|(Du)Bϱ|p2pE(u;Bϱ)p+2pχpE(u;Bϱ)p.

    Consider now

    κ:=cχE(u;Bϱ)+((ϱε3)mBϱ|f|mdx)1m(p1)andv0:=uκ,

    for ε3(0,1], which will be fixed later on. Applying (3.24) to the function v0 yields

    | Bϱ/2(x0)|Dv0|p2Dv0,Dφdx|(3.27)2,(3.29)cDφ(s+ω(s)1ε2+ε3).

    For any ε>0 and ϑ(0,1) and let δ be the one given by [17, Lemma 1.1]. Then, up to choosing s, ε2 and ε3 sufficiently small, we arrive at

    c(s+ω(s)1ε2+ε3)δDφp1.

    Then, Lemma 1.1 in [17] implies

    ( Bϱ/2|V(Dv0)V(Dh)|2ϑdx)1ϑcε Bϱ/2|Du|pdx(3.29),(3.27)2cεεp2,

    up to taking ε as small as needed. Now, denoting with h0:=hκ, we have that

    ( Bϱ/2|V(Du)V(Dh0)|2ϑdx)1ϑεεp2κp.

    Now, we choose ϑ:=(s)/2, with s being the exponent given by (2.8). Note that by the proof of (2.8) it actually follows that ϑ<1. Thus, choosing εεp2κpτ2n+4α1 (where α(0,1) is given by (3.34)) we arrive at

    ( Bϱ/2|V(Du)V(Dh0)|(s)dx)1(s)cτn+2α1.

    By Hölder's Inequality, we have that

    Bϱ/2|V(Du)V(Dh0)|2dx( Bϱ/2|V(Du)V(Dh0)|(s)dx)1(s)( Bϱ/2|V(Du)V(Dh0)|sdx)1s. (3.30)

    Hence, since by (2.3) V(z)|z|p, an application of estimates (2.8) and (3.29) now yields

    ( Bϱ/2|V(Du)|sdx)1sc( Bϱ/2|Du(Du)ϱ|p2dx)pp2+c|(Du)ϱ|pc Bϱ|Du|pdx+c(ϱmBϱ|f|mdx)pm(p1)+c|(Du)ϱ|pcE(u;Bϱ)p+c(ϱmBϱ|f|mdx)pm(p1), (3.31)

    with cc(data,χ).

    On the other hand, by classical properties of p-harmonic functions, we have that

    ( Bϱ/2|V(Dh0)|sdx)1sc Bϱ|Dh0|pdxc Bϱ|Du|pdxcE(u;Bϱ)p. (3.32)

    Hence, combining (3.30)–(3.32), we get that

    Bϱ/2|V(Du)V(Dh0)|2dxcτn+2α1E(u;Bϱ)p+cτn+2α1(ϱmBϱ|f|mdx)pm(p1). (3.33)

    Let us recall that, for any τ1(0,210), given the p-harmonic function h0 we have

    ˜E(h0;Bτ1ϱ)2cτ2α1κp,αα(n,N,p)(0,1). (3.34)

    Moreover, using Jensen's Inequality we can estimate the following difference as follows

    |(Du)τ1ϱ(Du)ϱ|( Bτ1ϱ|Du(Du)ϱ|pdx)1pτnp1( Bϱ|Du(Du)ϱ|pdx)1p(1.4),(3.27)2τnp1ε2.

    Thus, up to taking ε2 sufficiently small, by the triangular inequality, we obtain that

    12|(Du)τ1ϱ||(Du)ϱ|2|(Du)τ1ϱ|.

    Hence, (2.2) yield

    |V|(Du)τ1ϱ|()|2|V|(Du)ϱ|()|2,

    and

    |V((Du)τ1ϱ)V((Du)ϱ)|2|V|(Du)ϱ|((Du)ϱ(Du)τ1ϱ)|2.

    Then,

    E(u;Bτ1ϱ)p(2.5)c˜E(u;Bτ1ϱ)2(2.6)cBτ1ϱ|V(Du)V((Du)τ1ϱ)|2dxcτn1 Bϱ/2|V(Du)V(Dh0)|2dx+c Bτ1ϱ|V(Dh0)V((Dh0)τ1ϱ)|2dx(2.6)cτn1 Bϱ/2|V(Du)V(Dh0)|2dx+c ˜E(h0,Bτ1ϱ)(3.33),(3.34)cτ2α1E(u;Bϱ)p+c(ϱmBϱ|f|mdx)pm(p1),

    and the desired estimate (3.28) follows.

    This section is devoted to the proof of Theorem 1.1. First, we prove the following proposition.

    Proposition 4.1. Under assumptions (1.6)1,2,3, (1.7) and (1.10), let uW1,p(Ω,RN) be a local minimizer of functional (1.1). Then, there exists εε(data)>0 such that if the following condition

    E(Du;Br)+supϱr(ϱmBϱ|f|mdx)1m(p1)<ε, (4.1)

    is satisfied on BrΩ, for some ε(0,ε], then

    supϱrE(Du;Bϱ)<c3ε, (4.2)

    for c3c3(data)>0.

    Proof. For the sake of readability, since all balls considered in the proof are concentric to Br(x0), we will omit denoting the center.

    Let us start fixing an exponent ββ(α,p) such that

    0<β<min{β0,β1}=:βm, (4.3)

    where β0 and β1 are the exponents appearing in Propositions 3.1 and 3.2. Moreover, given the constant c0 and c1 from Propositions 3.1 and 3.2, choose ττ(data,β) such that

    (c0+c1)τβmβ14. (4.4)

    With the choice of τ0 as in (4.4) above, we can determine the constant ε0 and ε1 of Proposition 3.1. Now, we proceed applying Proposition 3.2 taking χε0 and τ1 as in (4.4) there. This determines the constant ε2 and c2. We consider a ball BrΩ such that

    E(Du;Br)<ε2, (4.5)

    and

    supϱrc2(ϱmBϱ|f|mdx)1m(p1)ε24, (4.6)

    where the constant c2:=c1+c0, with c0 appearing in (3.5) and c1 in (3.28). In particular, see that by (4.5) and (4.6) we are in the case when (4.1) does hold true.

    Now, we recall Proposition 3.2. Seeing that (3.27)2 is satisfied (being (4.5)) we only check whether (3.27)1 is verified too. If ε0|(Du)Br|E(Du;Br) is satisfied then we obtain from (3.28), with τ1τ in (4.4) that

    E(u;Bτr)τβ4E(u;Br)+c2(rmBr|f|m dx)1m(p1)τβ4E(u;Br)+supϱrc2(ϱmBϱ|f|m dx)1m(p1)τβ4E(u;Br)+ε24ε2, (4.7)

    where the last inequality follows from (4.5) and (4.6). If on the other hand it holds ε0|(Du)Br|E(Du;Br), by Proposition 3.1, then by (3.3) or (3.5) we eventually arrive at the same estimate (4.7).

    Iterating now the seam argument we arrive at

    E(Du;Bτjr)<ε2for any j0,

    and the estimate

    E(u;Bτj+1r)τβ4E(u;Bτjr)+c2((τjr)mBτjr|f|m dx)1m(p1),

    holds true. By the inequality above we have that for any k0

    E(u;Bτk+1r)τβ(k+1)4E(u;Br)+c2kj=0(τβ)jk((τjr)mBτjr|f|m dx)1m(p1)τβ(k+1)E(u;Br)+c2supϱr(ϱmBrr|f|mdx)1m(p1).

    Applying a standard interpolation argument we conclude that, for any tr, it holds

    E(Du,Bs)c3(sr)βE(Du,Br)+c3supϱr(ϱmBrr|f|mdx)1m(p1), (4.8)

    where c3c3(data). The desired estimate (4.2) now follows.

    Proof of Theorem 1.1. We proceed following the same argument used in [33, Theorem 1.5]. We star proving that, for any 1m<n and any OΩ, with positive measure, we have that

    fLm(O)(nnm)1/m|O|1/m1/nfLn,(O). (4.9)

    Indeed, fix ˉλ which will be chosen later on. Then, we have that

    fmLm(O)=mˉλ0λm|{xO:|f|>λ}|dλλ+mˉλλm|{xO:|f|>λ}|dλλ. (4.10)

    The first integral on the righthand side of (4.10) can be estimated in the following way

    ˉλ0λm|{xO:|f|>λ}|dλλˉλm|O|m.

    On the other hand, the second integral can be estimated recalling the definition of the Ln,(O)-norm. Indeed,

    ˉλλm|{xO:|f|>λ}|dλλfnLn,(O)ˉλdλλ1+nmfnLn,(O)(nm)ˉλnm.

    Hence, putting all the estimates above in (4.10), choosing ˉλ:=fLn,(O)/|O|1/n, we obtain (4.9).

    Now, recalling condition (1.2) we have that

    (ϱmBϱ|f|mdx)1/m(nnm)1/m|B1|1/nfLn,(Ω)(1.10)(4n/m|B1|)1/nfLn,(Ω)(1.2)ε,

    where ε is the one obtained in the proof of Proposition 4.1. From this it follows that, we can choose a radius ϱ1 such that

    supϱϱ1c2(ϱmBϱ(x)|f|mdx)1/m(p1)ε4c3. (4.11)

    We want to show that the set Ωu appearing in (1.3) can be characterized by

    Ωu:={x0Ω:Bϱ(x0)Ωwithϱϱ1:E(Du,Bϱ(x0))<ε/(4c3)},

    thus fixing ϱx0:=ϱ1 and εx0:=ε/(4c3). We first star noting that the the set Ωu defined in (1.4) is such that |ΩsetminusΩu|=0. Indeed, let us consider the set

    Lu:={x0Ω:lim infϱ0˜E(u;Bϱ(x0))2=0}, (4.12)

    which is such that |ΩsetminusLu|=0 by standard Lebesgue's Theory. Moreover, by (2.5) it follows that

    Lu:={x0Ω:lim infϱ0E(u;Bϱ(x0))=0},

    so that, LuΩu and we eventually obtained that |ΩsetminusΩu|=0. Now we show that Ωu is open. Let us fix x0Ωu and find a radius ϱx0ϱ1 such that

    E(Du,Bϱx0(x0))<ε4c3. (4.13)

    By absolute continuity of the functional E() we have that there exists an open neighbourhood O(x0) such that, for any xO(x0) it holds

    E(Du,Bϱx0(x))<ε4c3andBϱx0(x)Ω. (4.14)

    This prove that Ωu is open. Now let us start noting that (4.11) and (4.14) yield that condition (4.1) is satisfied with BrBϱx0(x). Hence, an application of Proposition 4.1 yields

    suptϱx0E(Du,Bt(x))<ε,

    for any xO(x0). Thus concluding the proof.

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

    The author is supported by INdAM Projects "Fenomeni non locali in problemi locali", CUP_E55F22000270001 and "Problemi non locali: teoria cinetica e non uniforme ellitticità", CUP_E53C220019320001, and also by the Project "Local vs Nonlocal: mixed type operators and nonuniform ellipticity", CUP_D91B21005370003.

    The author declares no conflict of interest.


    Acknowledgments



    Supported by the Department of Science and Technology, Government of India, New Delhi under the Inspire Faculty Award Scheme of AORC (IFA15/LSBM-154).

    Conflict of interest



    The authors declare no conflict of interest.

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