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Research article

An optimal resource allocation scheme for virtual machine placement of deploying enterprise applications into the cloud

  • The emergence of cloud computing can help enterprises reduce their hardware and software investment and save their own operation and maintenance costs, thus more and more enterprises deploy their applications into the cloud. Generally, components of enterprise applications are resided in virtual machines and then hosted by physical machines. In order to achieve the efficiency and utilization of physical machines, reasonable virtual machines placement becomes very important. In this paper we propose a scheme of resource allocation model for virtual machines placement and investigate it with convex optimization approach. We also present a heuristic algorithm to achieve the optimal resource allocation and discuss its equilibrium and stability by applying the asymptotic stability of the continuous dynamic system of Lyapunov stability theory. Finally, we give some numerical examples to illustrate the performance of the resource allocation scheme and confirm its convergence with a certain number of iterations.

    Citation: Wei Sun, Yan Wang, Shiyong Li. An optimal resource allocation scheme for virtual machine placement of deploying enterprise applications into the cloud[J]. AIMS Mathematics, 2020, 5(4): 3966-3989. doi: 10.3934/math.2020256

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  • The emergence of cloud computing can help enterprises reduce their hardware and software investment and save their own operation and maintenance costs, thus more and more enterprises deploy their applications into the cloud. Generally, components of enterprise applications are resided in virtual machines and then hosted by physical machines. In order to achieve the efficiency and utilization of physical machines, reasonable virtual machines placement becomes very important. In this paper we propose a scheme of resource allocation model for virtual machines placement and investigate it with convex optimization approach. We also present a heuristic algorithm to achieve the optimal resource allocation and discuss its equilibrium and stability by applying the asymptotic stability of the continuous dynamic system of Lyapunov stability theory. Finally, we give some numerical examples to illustrate the performance of the resource allocation scheme and confirm its convergence with a certain number of iterations.


    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

    (1.2)

    then there exists an open set \varOmega_u \subset \varOmega with \lvert { \varOmega \setminus \varOmega_u}\rvert = 0 such that

    \begin{equation} D u \in BMO_{{\operatorname{loc}}}( \varOmega_u; \mathbb{R}^{N\times n}). \end{equation} (1.3)

    Moreover, the set \varOmega_u can be characterized as follows

    \begin{eqnarray} \label{t1 2} \varOmega_u : = \left\{ x_0 \in \varOmega : \exists \varepsilon _{x_0}, \varrho_{x_0} > 0\right. \; \left.\mathit{\text{such that}} \ \mathscr{E}(u;B_ \varrho(x_0)) \leq\varepsilon _{x_0} \ \mathit{\text{for some}} \ \varrho \leq \varrho_{x_0} \right\}, \end{eqnarray}

    where \mathscr{E}(\cdot) is the usual excess functional defined as

    \begin{equation} \mathscr{E}(w,z_0;B_ \varrho(x_0)) : = \left( \ \rlap{-} \displaystyle {\int }_{B_ \varrho(x_0)} \lvert {z_0}\rvert^{p-2} \lvert { D w -z_0}\rvert^2 + \lvert { D w - z_0}\rvert^p \ {\,{{\rm{d}}}x} \right)^\frac{1}{p}. \end{equation} (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(\cdot) is a quasiconvex whenever

    \begin{align} \rlap{-} \displaystyle {\int }_{B_{1}(0)}F(z+D\varphi) {\,{{\rm{d}}}x} \geq F(z) \quad \mbox{holds for all} \ \ z\in \mathbb{R}^{N\times n}, \ \ \varphi\in C^{\infty}_{\rm c}(B_{1}(0), \mathbb{R}^{N}). \end{align} (1.5)

    Under power growth conditions, (1.5) is proven to be necessary and sufficient for the sequential weak lower semicontinuity on W^{1, p}(\varOmega; \mathbb{R}^N) ; 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(\cdot) 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 \mathcal{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., f\equiv 0 in (1.1). The first sharp \varepsilon -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 - \varDelta u = 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 C^{1} -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 \mbox{BMO} -regularity for minima of quasiconvex functionals expressed in terms of the limiting function space L(n, \infty) .

    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\colon \mathbb{R}^{N\times n}\to \mathbb{R} satisfies

    \begin{align} \begin{cases} \ F\in C^{2}_{{\operatorname{loc}}}( \mathbb{R}^{N\times n})\\ \ \varLambda^{-1}\lvert {z}\rvert^{p} \leq F(z) \leq \varLambda \lvert {z}\rvert^{p}\\ \ \lvert {\partial^{2}F(z)}\rvert \leq \varLambda \lvert {z}\rvert^{p-2}\\ \ \lvert {\partial^{2} F(z_{1})-\partial^{2}F(z_{2})}\rvert\le\mu\left( \frac{\lvert {z_{2}-z_{1}}\rvert}{\lvert {z_{2}}\rvert+\lvert {z_{1}}\rvert}\right) \left(\lvert {z_{1}}\rvert^{2}+\lvert {z_{2}}\rvert^{2}\right)^{\frac{p-2}{2}} \end{cases} \end{align} (1.6)

    for all z\in \mathbb{R}^{N\times n} , \varLambda \geq 1 being a positive absolute constant and \mu\colon [0, \infty)\to [0, 1] being a concave nondecreasing function with \mu(0) = 0 . In the rest of the paper we will always assume p\ge 2 . In order to derive meaningful regularity results, we need to update (1.5) to the stronger strict quasiconvexity condition

    \begin{align} \int_{B}\left[F(z+D\varphi)-F(z)\right]\ {\,{{\rm{d}}}x} \geq \lambda\int_{B}(\lvert {z}\rvert^{2}+\lvert {D\varphi}\rvert^{2})^{\frac{p-2}{2}}\lvert {D\varphi}\rvert^{2} {\,{{\rm{d}}}x}, \end{align} (1.7)

    holding for all z\in \mathbb{R}^{N\times n} and \varphi\in W^{1, p}_{0}(B, \mathbb{R}^{N}) , with \lambda being a positive, absolute constant. Furthermore, we allow the integrand F(\cdot) to be degenerate elliptic in the origin. More specifically, we assume that F(\cdot) features degeneracy of p -Laplacian type at the origin, i.e.,

    \begin{align} \left| \ \frac{\partial F(z)-\partial F(0)-\lvert {z}\rvert^{p-2}z}{\lvert {z}\rvert^{p-1}} \ \right|\to 0\, \qquad \text{as}\; \lvert {z}\rvert\to 0, \end{align} (1.8)

    which means that we can find a function \omega\colon (0, \infty)\to (0, \infty) such that

    \begin{eqnarray} \lvert {z}\rvert \leq\omega(s) \ \Longrightarrow \ \lvert {\partial F(z)-\partial F(0)-\lvert {z}\rvert^{p-2}z}\rvert \leq s\lvert {z}\rvert^{p-1}, \end{eqnarray} (1.9)

    for every z\in \mathbb{R}^{N\times n} and all s\in (0, \infty) . Moreover, the right-hand side term f\colon \varOmega\to \mathbb{R}^{N} in (1.1) verifies as minimal integrability condition the following

    \begin{eqnarray} f\in L^{m}( \varOmega, \mathbb{R}^{N})\quad \mbox{with} \ \ 2 > m > \begin{cases} \ 2n/(n+2)\quad &\mbox{if} \ \ n > 2,\\ \ 3/2\quad &\mbox{if} \ \ n = 2, \end{cases} \end{eqnarray} (1.10)

    which, being p \geq 2 , in turn implies that

    \begin{eqnarray} f\in W^{1,p}( \varOmega, \mathbb{R}^{N})^{*}\qquad \mbox{and}\qquad m' < 2^{*} \leq p^{*}. \end{eqnarray} (1.11)

    Here it is intended that, when p \geq n , 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

    \begin{eqnarray} \lvert {\partial F(z)}\rvert \leq c\lvert {z}\rvert^{p-1}, \end{eqnarray} (1.12)

    with c\equiv c(n, N, \varLambda, p) ; see for example [35, proof of Theorem 2.1]. Finally, (1.7) yields that for all z\in \mathbb{R}^{N\times n} , \xi\in \mathbb{R}^{N} , \zeta\in \mathbb{R}^{n} it is

    \begin{eqnarray} \partial^{2}F(z)\langle\xi\otimes \zeta,\xi\otimes \zeta\rangle \geq 2\lambda\lvert {z}\rvert^{p-2}\lvert {\xi}\rvert^{2}\lvert {\zeta}\rvert^{2}, \end{eqnarray} (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, \varOmega\subset {\mathbb R}^n is an open, bounded domain with Lipschitz boundary, and n \geq 2 . 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_*, \tilde 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 {\texttt{data}}: = \left(n, N, \lambda, \varLambda, p, \mu(\cdot), \omega(\cdot)\right) . By B_r(x_0): = \{x \in {\mathbb R}^n : |x-x_0| < r\} , we denote the open ball with radius r , centred at x_{0} ; when not necessary or clear from the context, we shall omit denoting the center, i.e., B_{r}(x_{0})\equiv B_{r} - this will happen, for instance, when dealing with concentric balls. For x_{0}\in \varOmega , we abbreviate d_{x_{0}}: = \min\left\{1, {\, {{\rm{dist}}}}(x_{0}, \partial \varOmega)\right\} . Moreover, with B \subset {\mathbb R}^{n} being a measurable set with bounded positive Lebesgue measure 0 < | B| < \infty , and a \colon B \to {\mathbb R}^{k} , k \geq 1 , being a measurable map, we denote

    (a)_{ B} \equiv \rlap{-} \displaystyle {\int }_{ B} a(x)\ {\,{{\rm{d}}}x} : = \frac{1}{| B|}\int_{B} a(x){\,{{\rm{d}}}x}.

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

    \begin{align} \left( \ \rlap{-} \displaystyle {\int }_{B}\lvert {a-(a)_{B}}\rvert^{t} {\,{{\rm{d}}}x}\right)^{1/t} \leq2\left( \ \rlap{-} \displaystyle {\int }_{B}\lvert {a-z}\rvert^{t} {\,{{\rm{d}}}x}\right)^{1/t} \end{align} (2.1)

    for all z\in \mathbb{R}^{N\times n} and any t \geq 1 . Finally, if t > 1 we will indicate its conjugate by t': = t/(t-1) and its Sobolev exponents as t^{*}: = nt/(n-t) if t < n or any number larger than one for t \geq n and t_{*}: = \max\left\{nt/(n+t), 1\right\} .

    When dealing with p -Laplacian type problems, we shall often use the auxiliary vector field V_{s}\colon {\mathbb R}^{N\times n} \to {\mathbb R}^{N\times n} , defined by

    \begin{align*} V_{s}(z): = (s^{2}+|z|^{2})^{(p-2)/4}z\qquad \text{with}\; p\in (1,\infty), \ \ s \geq 0, \ \ z\in \mathbb{R}^{N\times n}, \end{align*}

    incorporating the scaling features of the p -Laplacian. If s = 0 we simply write V_{s}(\cdot)\equiv V(\cdot) . A couple of useful related inequalities are

    \begin{align} \begin{cases} \ \lvert {V_{s}(z_{1})-V_{s}(z_{2})}\rvert\approx (s^{2}+\lvert {z_{1}}\rvert^{2}+\lvert {z_{2}}\rvert^{2})^{(p-2)/4}\lvert {z_{1}-z_{2}}\rvert,\\ \ \lvert {V_{s}(z_{1}+z_{2})}\rvert\lesssim \lvert {V_{s}(z_{1})}\rvert+\lvert {V_{s}(z_{2})}\rvert,\\ \ \lvert {V_{s_1}(z)}\rvert \approx \lvert {V_{s_2}(z)}\rvert, \ \mbox{if}\ \frac{1}{2}s_2 \leq s_1 \leq2 s_2,\\ \ \lvert {V(z_1)-V(z_2)}\rvert^2 \approx \lvert {V_{\lvert {z_1}\rvert}(z_1-z_2)}\rvert^2, \ \mbox{if}\ \frac{1}{2}\lvert {z_2}\rvert \leq\lvert {z_1}\rvert \leq2 \lvert {z_2}\rvert, \end{cases} \end{align} (2.2)

    and

    \begin{eqnarray} \lvert {V_{s}(z)}\rvert^{2}\approx s^{p-2}\lvert {z}\rvert^{2}+\lvert {z}\rvert^{p}\qquad \mbox{with} \ \ p \geq 2, \end{eqnarray} (2.3)

    where the constants implicit in " \lesssim ", " \approx " 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\in [0, 1] and z_{1}, z_{2}\in \mathbb{R}^{N\times n} be such that s+\lvert {z_{1}}\rvert+\lvert {z_{2}}\rvert > 0 . Then

    \begin{align*} \int_{0}^{1}\left[s^2+\lvert {z_{1}+y(z_{2}-z_{1})}\rvert^{2}\right]^{\frac{t}{2}} \ {\,{{\rm{d}}}y}\approx (s^2+\lvert {z_{1}}\rvert^{2}+\lvert {z_{2}}\rvert^{2})^{\frac{t}{2}}, \end{align*}

    with constants implicit in " \approx " 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\colon [ \varrho_{0}, \varrho_{1}]\to \mathbb{R} be a non-negative and bounded function, and let \theta \in (0, 1) , A, B, \gamma_{1}, \gamma_{2} \geq 0 be numbers. Assume that h(t) \leq\theta h(s)+A(s-t)^{-\gamma_{1}}+B(s-t)^{-\gamma_{2}} holds for all \varrho_{0} \leq t < s \leq \varrho_{1} . Then the following inequality holds h(\varrho_{0}) \leq c(\theta, \gamma_{1}, \gamma_{2})[A(\varrho_{1}- \varrho_{0})^{-\gamma_{1}}+B(\varrho_{1}- \varrho_{0})^{-\gamma_{2}}].

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

    \begin{equation} \widetilde{\mathscr{E}}(w,z_0;B_ \varrho(x_0)) : = \left( \ \rlap{-} \displaystyle {\int }_{B_ \varrho(x_0)} \lvert {V( D w)-z_0}\rvert^2 {\,{{\rm{d}}}x} \right)^\frac{1}{2}. \end{equation} (2.4)

    In the particular case z_0 = (D w)_{B_ \varrho(x_0)} ( z_0 = (V(D w))_{B_ \varrho(x_0)} , resp.) we shall simply write \mathscr{E}(w, (D w)_{B_ \varrho(x_0)}; B_ \varrho (x_0)) \equiv \mathscr{E}(w; B_ \varrho(x_0)) ( \widetilde{\mathscr{E}}(w, (V(D w))_{B_ \varrho(x_0)}; B_ \varrho(x_0))\equiv \widetilde{\mathscr{E}}(w; B_ \varrho(x_0)) , resp.). A simple computation shows that

    \begin{equation} \mathscr{E}(w;B_ \varrho(x_0))^{p/2} \approx \widetilde{\mathscr{E}}(w;B_ \varrho(x_0)). \end{equation} (2.5)

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

    \begin{equation} \widetilde{\mathscr{E}}(w;B_ \varrho(x_0)) \approx \widetilde{\mathscr{E}}(w, V(( D w)_{B_ \varrho(x_0)});B_ \varrho(x_0)). \end{equation} (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 u\in W^{1, p}(\varOmega, \mathbb{R}^{N}) be a local minimizer of functional (1.1).

    For every ball B_{ \varrho}(x_{0})\Subset \varOmega and any u_{0}\in \mathbb{R}^{N} , z_{0}\in \mathbb{R}^{N\times n}\setminus \{0\} it holds that

    \begin{eqnarray} \mathscr{E}(u,z_0;B_{ \varrho/2}(x_0))^p & \leq& c\, \rlap{-} \displaystyle {\int }_{B_{ \varrho}(x_{0})}\lvert {z_{0}}\rvert^{p-2}{\left|{\frac{u-\ell}{ \varrho}}\right|}^{2}+{\left|{\frac{u-\ell}{ \varrho}}\right|}^{p} {\,{{\rm{d}}}x} \\ &&+\,\frac{c}{\lvert {z_{0}}\rvert^{p-2}}\left( \varrho^{m}\rlap{-} \displaystyle {\int }_{B_{ \varrho}(x_{0})}\lvert {f}\rvert^{m} \ {\,{{\rm{d}}}x}\right)^{\frac{2}{m}}, \end{eqnarray} (2.7)

    where \mathscr{E}(\cdot) is defined in (1.4), \ell(x): = u_{0}+\langle z_0, x-x_{0}\rangle and c\equiv c(n, N, \lambda, \varLambda, p) .

    There exists an higher integrability exponent p_{2}\equiv p_{2}(n, N, \lambda, \varLambda, p) > p such that Du\in L^{p_{2}}_{{\operatorname{loc}}}(\varOmega, \mathbb{R}^{N\times n}) and the reverse Hölder inequality

    \begin{eqnarray} && \left(\ \rlap{-} \displaystyle {\int }_{B_{ \varrho/2}(x_{0})}\lvert {Du-(Du)_{B_{ \varrho}(x_{0})}}\rvert^{p_{2}} {\,{{\rm{d}}}x}\right)^{\frac{1}{p_{2}}}\\ &&\qquad \leq c\left( \ \rlap{-} \displaystyle {\int }_{B_{ \varrho}(x_{0})}\lvert {Du}\rvert^{p} {\,{{\rm{d}}}x}\right)^{\frac{1}{p}}+c\left( \varrho^{m}\rlap{-} \displaystyle {\int }_{B_{ \varrho}(x_{0})}\lvert {f}\rvert^{m} {\,{{\rm{d}}}x}\right)^{\frac{1}{m(p-1)}}, \end{eqnarray} (2.8)

    is verified for all balls B_{ \varrho}(x_{0})\Subset \varOmega with c\equiv c(n, N, \lambda, \varLambda, 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 \varrho/2 \leq\tau_{1} < \tau_{2} \leq \varrho , a cut-off function \eta\in C^{1}_{c}(B_{\tau_{2}}(x_{0})) such that \mathbb{1}_{B_{\tau_{1}}(x_{0})} \leq\eta \leq \mathbb{1}_{B_{\tau_{2}}(x_{0})} and \lvert {D\eta}\rvert\lesssim (\tau_{2}-\tau_{1})^{-1} . Set \varphi_{1}: = \eta(u-\ell) , \varphi_{2}: = (1-\eta)(u-\ell) and use (1.7) and the equivalence in (2.2) _{1} to estimate

    \begin{eqnarray} c \int_{B_{\tau_{2}}(x_{0})} \lvert {V_{\lvert {z_{0}}\rvert}(D\varphi_{1})}\rvert^{2} \ {\,{{\rm{d}}}x} & \leq & \int_{B_{\tau_{2}}(x_{0})} [F(z_{0}+D \varphi_{1})-F(z_{0})] {\,{{\rm{d}}}x} \\ & = & \int_{B_{\tau_{2}}(x_{0})}[F(Du-D \varphi_{2})-F(Du)] {\,{{\rm{d}}}x} \\ && +\int_{B_{\tau_{2}}(x_{0})}[F(Du)-F(Du-D \varphi_{1})] {\,{{\rm{d}}}x}\\ && +\int_{B_{\tau_{2}}(x_{0})} [F(z_{0}+D \varphi_{2})-F(z_{0})] {\,{{\rm{d}}}x} = : \mbox{I}_1 +\mbox{I}_2+ \mbox{I}_3, \end{eqnarray} (2.9)

    where we have used the simple relation D \varphi_{1} + D \varphi_{2} = Du -z_{0} . Terms \mbox{I}_{1} and \mbox{I}_{3} can be controlled as done in [19, Proposition 2]; indeed we have

    \begin{eqnarray} \mbox{I}_{1}+\mbox{I}_{3}& \leq& c\int_{B_{\tau_{2}}(x_{0})\setminus B_{\tau_{1}}(x_{0})}\lvert {V_{\lvert {z_{0}}\rvert}(D\varphi_{2})}\rvert^{2} {\,{{\rm{d}}}x}+c\int_{B_{\tau_{2}}(x_{0})\setminus B_{\tau_{1}}(x_{0})}\lvert {V_{\lvert {z_{0}}\rvert}(Du-z_{0})}\rvert^{2} {\,{{\rm{d}}}x} \\ &\stackrel{(2.2)_{2}}{ \leq}&c\int_{B_{\tau_{2}}(x_{0})\setminus B_{\tau_{1}}(x_{0})}\lvert {V_{\lvert {z_{0}}\rvert}(Du-z_{0})}\rvert^{2}+\left|\ V_{\lvert {z_{0}}\rvert}\left(\frac{u-\ell}{\tau_{2}-\tau_{1}}\right)\ \right|^{2} {\,{{\rm{d}}}x}, \end{eqnarray} (2.10)

    for c\equiv c(n, N, \lambda, \varLambda, p) . Concerning term \mbox{I}_{2} , we exploit (1.10), the fact that \varphi_{1}\in W^{1, p}_{0}(B_{\tau_{2}}(x_{0}), \mathbb{R}^{N}) and apply Sobolev-Poincaré inequality to get

    \begin{eqnarray} \mbox{I}_{2}& \leq&\lvert {B_{\tau_{2}}(x_{0})}\rvert\left(\tau_{2}^{m}\rlap{-} \displaystyle {\int }_{B_{\tau_{2}}(x_{0})}\lvert {f}\rvert^{m}\ {\,{{\rm{d}}}x}\right)^{1/m}\left(\tau_{2}^{-m'}\rlap{-} \displaystyle {\int }_{B_{\tau_{2}}(x_{0})}\lvert {\varphi_{1}}\rvert^{m'}\ {\,{{\rm{d}}}x}\right)^{\frac{1}{m'}} \\ & \leq&\lvert {B_{\tau_{2}}(x_{0})}\rvert\left(\tau_{2}^{m}\rlap{-} \displaystyle {\int }_{B_{\tau_{2}}(x_{0})}\lvert {f}\rvert^{m}\ {\,{{\rm{d}}}x}\right)^{1/m}\left( \ \rlap{-} \displaystyle {\int }_{B_{\tau_{2}}(x_{0})}\left| \ \frac{\varphi_{1}}{\tau_{2}} \ \right|^{2^{*}}\ {\,{{\rm{d}}}x}\right)^{\frac{1}{2^{*}}} \\ & \leq&\lvert {B_{\tau_{2}}(x_{0})}\rvert\left(\tau_{2}^{m}\rlap{-} \displaystyle {\int }_{B_{\tau_{2}}(x_{0})}\lvert {f}\rvert^{m} {\,{{\rm{d}}}x}\right)^{1/m}\left( \ \rlap{-} \displaystyle {\int }_{B_{\tau_{2}}(x_{0})}\lvert {D\varphi_{1}}\rvert^{2} {\,{{\rm{d}}}x}\right)^{\frac{1}{2}} \\ & \leq&\varepsilon\int_{B_{\tau_{2}}(x_{0})}\lvert {V_{\lvert {z_{0}}\rvert}(D\varphi_{1})}\rvert^{2} {\,{{\rm{d}}}x}+\frac{c\lvert {B_{ \varrho}(x_{0})}\rvert}{\varepsilon\lvert {z_{0}}\rvert^{p-2}}\left( \varrho^{m}\rlap{-} \displaystyle {\int }_{B_{ \varrho}(x_{0})}\lvert {f}\rvert^{m} {\,{{\rm{d}}}x}\right)^{\frac{2}{m}}, \end{eqnarray} (2.11)

    where c\equiv c(n, N, m) and we also used that \varrho/2 \leq\tau_{2} \leq \varrho . Merging the content of the two above displays, recalling that \eta\equiv 1 on B_{\tau_{1}}(x_{0}) and choosing \varepsilon > 0 sufficiently small, we obtain

    \begin{eqnarray*} \int_{B_{\tau_{1}}(x_{0})}\lvert {V_{\lvert {z_{0}}\rvert}(Du-z_{0})}\rvert^{2} {\,{{\rm{d}}}x}& \leq&c\int_{B_{\tau_{2}}(x_{0})\setminus B_{\tau_{1}}(x_{0})}\lvert {V_{\lvert {z_{0}}\rvert}(Du-z_{0})}\rvert^{2}+\left| \ V_{\lvert {z_{0}}\rvert}\left(\frac{u-\ell}{\tau_{2}-\tau_{1}}\right)\ \right|^{2} {\,{{\rm{d}}}x} \nonumber \\ &&+\frac{c\lvert {B_{ \varrho}(x_{0})}\rvert}{\lvert {z_{0}}\rvert^{p-2}}\left( \varrho^{m}\rlap{-} \displaystyle {\int }_{B_{ \varrho}(x_{0})}\lvert {f}\rvert^{m} {\,{{\rm{d}}}x}\right)^{\frac{2}{m}}, \end{eqnarray*}

    with c\equiv c(n, N, \lambda, \varLambda, 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 \mathcal{F}(\cdot) on B_{ \varrho}(x_{0}) , setting f_{ \varrho}(x): = \varrho f(x_{0}+ \varrho x) , the map u_{ \varrho}(x): = \varrho^{-1}u(x_{0}+ \varrho x) is a local minimizer on B_{1}(0) of an integral with the same integrand appearing in (1.1) satisfying (1.6)_{1, 2, 3} and f_{ \varrho} replacing f . This means that (2.10) still holds for all balls B_{\sigma/2}(\tilde{{x}})\subseteq B_{\tau_{1}}(\tilde{{x}})\subset B_{\tau_{2}}(\tilde{{x}})\subseteq B_{\sigma}(\tilde{{x}})\Subset B_{1}(0) , with \tilde{{x}}\in B_{1}(0) being any point, in particular it remains true if \lvert {z_{0}}\rvert = 0 , while condition \lvert {z_{0}}\rvert\not = 0 was needed only in the estimate of term \mbox{I}_{2} in (2.11), that now requires some change. So, in the definition of the affine map \ell we choose z_{0} = 0 , u_{0} = (u_{ \varrho})_{B_{\sigma}(\tilde{{x}})} and rearrange estimates (2.10) and (2.11) as:

    \mbox{I}_{1}+\mbox{I}_{3}\stackrel{(2.3)}{ \leq} c\int_{B_{\tau_{2}}(\tilde{{x}})\setminus B_{\tau_{1}}(\tilde{{x}})}\lvert {Du_{ \varrho}}\rvert^{p} +\left| \ \frac{u_{ \varrho}-(u_{ \varrho})_{B_{\sigma}(\tilde{{x}})}}{\tau_{2}-\tau_{1}} \ \right|^{p} {\,{{\rm{d}}}x},

    and, recalling that \varphi_{1}\in W^{1, p}_{0}(B_{\tau_{2}}(\tilde{{x}}), \mathbb{R}^{N}) , via Sobolev Poincaré, Hölder and Young inequalities and (1.11) _{2} , we estimate

    \begin{eqnarray*} \mbox{I}_2 & \leq & {\left|{B_{\tau_2}(\tilde{{x}})}\right|} \left(\tau_2^{(p^*)'} \rlap{-} \displaystyle {\int }_{B_{\tau_2}(\tilde{{x}})}{\left|{f_{ \varrho}}\right|}^{(p^*)'} \ {\,{{\rm{d}}}x}\right)^{\frac{1}{(p^*)'}} \left(\tau_2^{-p^*} \rlap{-} \displaystyle {\int }_{B_{\tau_2}(\tilde{{x}})}{\left|{ \varphi_1}\right|}^{p^*}\ {\,{{\rm{d}}}x} \right)^\frac{1}{p^*}\\ & \leq & c{\left|{B_{\tau_2}(\tilde{{x}})}\right|} \left(\tau_2^{(p^*)'} \rlap{-} \displaystyle {\int }_{B_{\tau_2}(\tilde{{x}})}{\left|{f_{ \varrho}}\right|}^{(p^*)'} {\,{{\rm{d}}}x}\right)^{\frac{1}{(p^*)'}} \left( \ \rlap{-} \displaystyle {\int }_{B_{\tau_2}(\tilde{{x}})}{\left|{ D \varphi_1}\right|}^{p} {\,{{\rm{d}}}x} \right)^\frac{1}{p}\\ & \leq & \frac{c\lvert {B_{\sigma}(\tilde{{x}})}\rvert}{\varepsilon^{1/(p-1)}}\left(\sigma^{(p^*)'} \rlap{-} \displaystyle {\int }_{B_\sigma(\tilde{{x}})}{\left|{f_{ \varrho}}\right|}^{(p^*)'} {\,{{\rm{d}}}x}\right)^{\frac{p}{(p^*)'(p-1)}}+ \varepsilon \int_{B_{\tau_2}(\tilde{{x}})}{\left|{ D \varphi_1}\right|}^{p} {\,{{\rm{d}}}x}, \end{eqnarray*}

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

    \begin{eqnarray} \rlap{-} \displaystyle {\int }_{B_{\sigma/2}(\tilde{{x}})}\lvert {Du_{ \varrho}}\rvert^{p} \ {\,{{\rm{d}}}x} \leq c \ \rlap{-} \displaystyle {\int }_{B_{\sigma}(\tilde{{x}})}\left| \ \frac{u_{ \varrho}-(u_{ \varrho})_{B_{\sigma}(\tilde{{x}})}}{\sigma} \ \right|^{p} \ {\,{{\rm{d}}}x}+c\left(\sigma^{(p^*)'} \rlap{-} \displaystyle {\int }_{B_\sigma(\tilde{{x}})}{\left|{f_{ \varrho}}\right|}^{(p^*)'} {\,{{\rm{d}}}x}\right)^{\frac{p}{(p^*)'(p-1)}}, \end{eqnarray} (2.12)

    for c\equiv c(n, N, \varLambda, \lambda, p) . Notice that

    \begin{align} n\left(\frac{p}{(p^*)'(p-1)}-1\right) \leq\frac{p}{p-1}, \end{align} (2.13)

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

    \begin{eqnarray*} && \left(\sigma^{(p^*)'} \rlap{-} \displaystyle {\int }_{B_\sigma(\tilde{{x}})}{\left|{f_{ \varrho}}\right|}^{(p^*)'} {\,{{\rm{d}}}x}\right)^{\frac{p}{(p^*)'(p-1)}}\notag\\ &&\quad \leq\sigma^{\frac{p}{p-1}-n\left(\frac{p}{(p^{*})'(p-1)}-1\right)}\left( \ \rlap{-} \displaystyle {\int }_{B_{1}(0)}\lvert {f_{ \varrho}}\rvert^{(p^{*})'} \ {\,{{\rm{d}}}x}\right)^{\frac{p}{(p^{*})'(p-1)}-1} \rlap{-} \displaystyle {\int }_{B_{\sigma}(\tilde{{x}})}\lvert {f_{ \varrho}}\rvert^{(p^{*})'} {\,{{\rm{d}}}x}\nonumber \\ &&\quad\stackrel{(2.13)}{ \leq}\left( \ \rlap{-} \displaystyle {\int }_{B_{1}(0)}\lvert {f_{ \varrho}}\rvert^{(p^{*})'} \ {\,{{\rm{d}}}x}\right)^{\frac{p}{(p^{*})'(p-1)}-1}\rlap{-} \displaystyle {\int }_{B_{\sigma}(\tilde{{x}})}\lvert {f_{ \varrho}}\rvert^{(p^{*})'} \ {\,{{\rm{d}}}x}\nonumber \\ &&\quad = :\rlap{-} \displaystyle {\int }_{B_{\sigma}(\tilde{{x}})}\lvert {\mathfrak{K}_{ \varrho}f_{ \varrho}}\rvert^{(p^{*})'} \ {\,{{\rm{d}}}x}, \end{eqnarray*}

    where we set

    \mathfrak{K}_{ \varrho}^{(p^{*})'}: = \lvert {B_{1}(0)}\rvert^{1-\frac{p}{(p^{*})'(p-1)}}\lVert {f_{ \varrho}} \rVert^{\frac{p}{p-1}-(p^{*})'}_{L^{(p^{*})'}(B_{1}(0))}.

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

    \begin{eqnarray*} \rlap{-} \displaystyle {\int }_{B_{\sigma/2}(\tilde{{x}})}\lvert {Du_{ \varrho}}\rvert^{p} {\,{{\rm{d}}}x} \leq c\left( \ \rlap{-} \displaystyle {\int }_{B_{\sigma}(\tilde{{x}})}\lvert {Du_{ \varrho}}\rvert^{p_{*}} \ {\,{{\rm{d}}}x}\right)^{\frac{p}{p_{*}}}+c \ \rlap{-} \displaystyle {\int }_{B_{\sigma}(\tilde{{x}})}\lvert {\mathfrak{K}_{ \varrho}f_{ \varrho}}\rvert^{(p^{*})'} \ {\,{{\rm{d}}}x}, \end{eqnarray*}

    with c\equiv c(n, N, \varLambda, \lambda, p) . Now we can apply a variant of Gehring lemma [24, Corollary 6.1] to determine a higher integrability exponent \mathfrak{s}\equiv \mathfrak{s}(n, N, \varLambda, \lambda, p) such that 1 < \mathfrak{s} \leq m/(p^{*})' and

    \begin{eqnarray} \label{4} \left( \ \rlap{-} \displaystyle {\int }_{B_{\sigma/2}(\tilde{{x}})}\lvert {Du_{ \varrho}}\rvert^{\mathfrak{s}p} {\,{{\rm{d}}}x}\right)^{\frac{1}{\mathfrak{s}p}} \leq c\left( \ \rlap{-} \displaystyle {\int }_{B_{\sigma}(\tilde{{x}})}\lvert {Du_{ \varrho}}\rvert^{p}{\,{{\rm{d}}}x}\right)^{\frac{1}{p}} +c\mathfrak{K}_{ \varrho}^{(p^{*})'/p}\left( \ \rlap{-} \displaystyle {\int }_{B_{\sigma}(\tilde{{x}})}\lvert {f_{ \varrho}}\rvert^{\mathfrak{s}(p^{*})'} \ {\,{{\rm{d}}}x}\right)^{\frac{1}{\mathfrak{s}p}} \end{eqnarray}

    for c\equiv c(n, N, \varLambda, \lambda, p) . Next, notice that

    \begin{align*} \mathfrak{K}_{ \varrho}^{(p^{*})'/p} = \left( \ \rlap{-} \displaystyle {\int }_{B_{1}(0)}\lvert {f_{ \varrho}}\rvert^{(p^{*})'} {\,{{\rm{d}}}x}\right)^{\frac{1}{(p^{*})'(p-1)}-\frac{1}{p}} \leq\left( \ \rlap{-} \displaystyle {\int }_{B_{1}(0)}\lvert {f_{ \varrho}}\rvert^{\mathfrak{s}(p^{*})'} {\,{{\rm{d}}}x}\right)^{\frac{1}{\mathfrak{s}(p^{*})'(p-1)}-\frac{1}{\mathfrak{s}p}}, \end{align*}

    so plugging this last inequality in (2.14) and recalling that \mathfrak{s}(p^{*})' \leq m , we obtain

    \begin{align*} \left( \ \rlap{-} \displaystyle {\int }_{B_{\sigma/2}(\tilde{{x}})}\lvert {Du_{ \varrho}}\rvert^{\mathfrak{s}p} {\,{{\rm{d}}}x}\right)^{\frac{1}{\mathfrak{s}p}} \leq c\left( \ \rlap{-} \displaystyle {\int }_{B_{\sigma}(\tilde{{x}})}\lvert {Du_{ \varrho}}\rvert^{p}{\,{{\rm{d}}}x}\right)^{\frac{1}{p}}+c\left( \ \rlap{-} \displaystyle {\int }_{B_{\sigma}(\tilde{{x}})}\lvert {f_{ \varrho}}\rvert^{m} \ {\,{{\rm{d}}}x}\right)^{\frac{1}{m(p-1)}}. \end{align*}

    Setting p_{2}: = \mathfrak{s}p > p above and recalling that \tilde{{x}}\in B_{1}(0) is arbitrary, we can fix \tilde{{x}} = 0 , scale back to B_{ \varrho}(x_{0}) 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 \mathscr{E}(u; B_ \varrho(x_0)) 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 u\in W^{1, p}(\varOmega, \mathbb{R}^{N}) be a local minimizer of functional (1.1). Then, for \tau_0 \in (0, 2^{-10}) , there exists \varepsilon _0 \equiv \varepsilon _0(\mathit{{\texttt{data}}}, \tau_0) \in (0, 1) and \varepsilon _1 \equiv \varepsilon _1(\mathit{{\texttt{data}}}, \tau_0) \in (0, 1) such that the following implications hold true.

    If the conditions

    \begin{equation} \mathscr{E}(u;B_ \varrho(x_0)) \leq\varepsilon _0 \lvert {( D u )_{B_ \varrho (x_0)}}\rvert, \end{equation} (3.1)

    and

    \begin{equation} \left( \varrho^m \rlap{-} \displaystyle {\int }_{B_{ \varrho}(x_{0})}\lvert {f}\rvert^m \ {\,{{\rm{d}}}x} \right)^\frac{1}{m} \leq \varepsilon _1 \lvert {( D u )_{B_ \varrho (x_0)}}\rvert^\frac{p-2}{2} \mathscr{E}(u; B_ \varrho(x_0))^\frac{p}{2}, \end{equation} (3.2)

    are verified on B_ \varrho(x_0) , then it holds that

    \begin{equation} \mathscr{E}(u; B_{\tau_0 \varrho}(x_0)) \leq c_0 \tau_0^{\beta_0} \mathscr{E}(u;B_ \varrho(x_0)), \end{equation} (3.3)

    for all \beta_0 \in (0, 2/p) , with c_0 \equiv c_0(\mathit{{\texttt{data}}}) > 0 .

    If condition (3.1) holds true and

    \begin{equation} \left( \varrho^m \rlap{-} \displaystyle {\int }_{B_{ \varrho}(x_{0})}\lvert {f}\rvert^m \ {\,{{\rm{d}}}x} \right)^\frac{1}{m} > \varepsilon _1 \lvert {( D u )_{B_ \varrho (x_0)}}\rvert^\frac{p-2}{2} \mathscr{E}(u; B_ \varrho(x_0))^\frac{p}{2}, \end{equation} (3.4)

    is satisfied on B_ \varrho(x_0) , then

    \begin{equation} \mathscr{E}(u; B_{\tau_0 \varrho}(x_0)) \leq c_0 \left( \varrho^m \rlap{-} \displaystyle {\int }_{B_{ \varrho}(x_{0})}\lvert {f}\rvert^m \ {\,{{\rm{d}}}x} \right)^\frac{1}{m(p-1)}, \end{equation} (3.5)

    for c_0 \equiv c_0(\mathit{{\texttt{data}}}) > 0 .

    Proof of Proposition 3.1. For the sake of readability, since all balls considered here are concentric to B_ \varrho(x_0) , we will omit denoting the center. Moreover, we will adopt the following notation (D u)_{B_ \varsigma (x_0)} \equiv (D u)_ \varsigma and, for all \varphi \in C^\infty_c(B_ \varrho; \mathbb{R}^N) , we will denote {\left\|{ D \varphi}\right\|}_{L^\infty(B_ \varrho)} \equiv {\left\|{ D \varphi}\right\|}_\infty . We spilt the proof in two steps.

    Step 1: proof of (3.3).

    With no loss of generality we can assume that \mathscr{E}(u; B_ \varrho) > 0 , which clearly implies, thanks to (3.1), that \lvert {(D u)_ \varrho}\rvert > 0 .

    We begin proving that condition (3.1) implies that

    \begin{equation} \rlap{-} \displaystyle {\int }_{B_ \varrho} \lvert { D u}\rvert^p {\,{{\rm{d}}}x} \leq c \lvert {( D u)_ \varrho}\rvert^p, \end{equation} (3.6)

    for a constant c \equiv c(p, \varepsilon _0) > 0 . Indeed,

    \begin{eqnarray*} \rlap{-} \displaystyle {\int }_{B_ \varrho} \lvert { D u}\rvert^p {\,{{\rm{d}}}x} & \leq& c \ \rlap{-} \displaystyle {\int }_{B_ \varrho} \lvert { D u - ( D u)_ \varrho}\rvert^p {\,{{\rm{d}}}x} + c \lvert {( D u)_ \varrho}\rvert^p\\ &\stackrel{(1.4)}{ \leq}& c \, \mathscr{E}(u;B_ \varrho)^p + c\lvert {( D u)_ \varrho}\rvert^p\\ &\stackrel{(3.1)}{ \leq}& c(\varepsilon _0^p+1) \lvert {( D u)_ \varrho}\rvert^p, \end{eqnarray*}

    and (3.6) follows.

    Consider now

    \begin{equation} B_ \varrho \ni x \mapsto u_0(x) : = \frac{\lvert {( D u )_ \varrho}\rvert^\frac{p-2}{2}\big(u(x)-(u )_ \varrho - \langle ( D u )_ \varrho,x-x_0\rangle\big)}{\mathscr{E}(u; B_ \varrho)^{p/2}}, \end{equation} (3.7)

    and

    d: = \left(\frac{\mathscr{E}(u;B_ \varrho)}{\lvert {( D u)_ \varrho}\rvert}\right)^\frac{p}{2}.

    Let us note that we have

    \begin{eqnarray*} && \rlap{-} \displaystyle {\int }_{B_ \varrho}\lvert { D u_0}\rvert^2 {\,{{\rm{d}}}x} + d^{p-2}\rlap{-} \displaystyle {\int }_{B_ \varrho}\lvert { D u_0}\rvert^p {\,{{\rm{d}}}x} \\ && \qquad \leq \frac{\lvert {( D u)_ \varrho}\rvert^{p-2}}{\mathscr{E}(u;B_ \varrho)^p}\rlap{-} \displaystyle {\int }_{B_ \varrho}\lvert { D u-( D u)_ \varrho}\rvert^2 {\,{{\rm{d}}}x} \\ &&\qquad\qquad + \left(\frac{\mathscr{E}(u;B_ \varrho)}{\lvert {( D u)_ \varrho}\rvert}\right)^\frac{p(p-2)}{2}\frac{\lvert {( D u)_ \varrho}\rvert^\frac{p(p-2)}{2}}{\mathscr{E}(u;B_ \varrho)^\frac{p^2}{2}}\rlap{-} \displaystyle {\int }_{B_ \varrho}\lvert { D u-( D u)_ \varrho}\rvert^p {\,{{\rm{d}}}x}\\ && \qquad \leq \frac{1}{\mathscr{E}(u;B_ \varrho)^p}\rlap{-} \displaystyle {\int }_{B_ \varrho}\lvert {( D u)_ \varrho}\rvert^{p-2}\lvert { D u-( D u)_ \varrho}\rvert^2 {\,{{\rm{d}}}x} \\ &&\qquad\quad + \frac{1}{\mathscr{E}(u;B_ \varrho)^p}\rlap{-} \displaystyle {\int }_{B_ \varrho}\lvert { D u-( D u)_ \varrho}\rvert^p {\,{{\rm{d}}}x} \leq 1. \end{eqnarray*}

    Since \lvert {(D u)_ \varrho}\rvert > 0 we have that the hypothesis of [12, Lemma 3.2] are satisfied with

    \begin{equation} \mathscr{A} : = \partial^2 F(( D u )_ \varrho)\lvert {( D u )_ \varrho}\rvert^{2-p}. \end{equation} (3.8)

    Then,

    \begin{eqnarray*} {\left|{ \ \rlap{-} \displaystyle {\int }_{B_ \varrho} \mathscr{A} \langle D u_0, D \varphi \rangle \ {\,{{\rm{d}}}x}}\right|} & \leq& \frac{c {\left\|{ D \varphi}\right\|}_\infty \lvert {( D u )_ \varrho}\rvert^\frac{2-p}{2}}{\mathscr{E}(u;B_ \varrho)^\frac{p}{2}} \left( \varrho^m \rlap{-} \displaystyle {\int }_{B_ \varrho} \lvert {f}\rvert^m \ {\,{{\rm{d}}}x}\right)^\frac{1}{m}\\ && + c{\left\|{ D \varphi}\right\|}_\infty \mu \left(\frac{\mathscr{E}(u;B_ \varrho)}{\lvert {( D u)_ \varrho}\rvert} \right)^\frac{1}{p} \left[1 + \left(\frac{\mathscr{E}(u;B_ \varrho)}{\lvert {( D u)_ \varrho}\rvert}\right)^\frac{p-2}{2} \right]\\ &\stackrel{(3.1),(3.2)}{ \leq}& c \varepsilon _1 {\left\|{ D \varphi}\right\|}_\infty +c {\left\|{ D \varphi}\right\|}_\infty \mu(\varepsilon _0)^\frac{1}{p}\big[1+ \varepsilon _0^\frac{p-2}{2} \big]. \end{eqnarray*}

    Fix \varepsilon > 0 and let \delta \equiv \delta({\texttt{data}}, \varepsilon ) > 0 be the one given by [33, Lemma 2.4] and choose \varepsilon _0 and \varepsilon _1 sufficiently small such that

    \begin{equation} c \, \varepsilon _1 + c \mu(\varepsilon _0)^\frac{1}{p}\big[1+ \varepsilon _0^\frac{p-2}{2} \big] \leq \delta . \end{equation} (3.9)

    With this choice of \varepsilon _0 and \varepsilon _1 it follows that u_0 is almost \mathscr{A} -harmonic on B_ \varrho , in the sense that

    {\left|{ \ \rlap{-} \displaystyle {\int }_{B_ \varrho}\mathscr{A}\langle D u_0, D \varphi \rangle \ {\,{{\rm{d}}}x}}\right|} \leq\delta {\left\|{ D \varphi}\right\|}_\infty,

    with \mathscr{A} as in (3.8). Hence, by [33, Lemma 2.4] we obtain that there exists h_0 \in W^{1, 2}(B_ \varrho; \mathbb{R}^N) which is \mathscr{A} -harmonic, i.e.,

    \int_{B_ \varrho}\mathscr{A}\langle D h_0, D \varphi \rangle \ {\,{{\rm{d}}}x} = 0 \quad \mbox{for all } \varphi \in C^\infty_c(B_ \varrho; \mathbb{R}^N),

    such that

    \begin{equation} \rlap{-} \displaystyle {\int }_{B_{3 \varrho/4}}\lvert { D h_0}\rvert^2 {\,{{\rm{d}}}x} + d^{p-2}\rlap{-} \displaystyle {\int }_{B_{3 \varrho/4}}\lvert { D h_0}\rvert^p {\,{{\rm{d}}}x} \leq 8^{2np}, \end{equation} (3.10)

    and

    \begin{equation} \rlap{-} \displaystyle {\int }_{B_{3 \varrho/4}}{\left|{\frac{u_0-h_0}{ \varrho}}\right|}^2 + d^{p-2}{\left|{\frac{u_0-h_0}{ \varrho}}\right|}^p {\,{{\rm{d}}}x} \leq \varepsilon . \end{equation} (3.11)

    We choose now \tau_0\in (0, 2^{-10}) , which will be fixed later on, and estimate

    \begin{eqnarray} && \rlap{-} \displaystyle {\int }_{B_{2\tau_0 \varrho}}{\left|{\frac{u_0(x)-h_0(x_{0})-\langle Dh_0(x_{0}),x-x_0\rangle}{\tau_0 \varrho}}\right|}^2 {\,{{\rm{d}}}x} \\ && \quad \leq c \,\rlap{-} \displaystyle {\int }_{B_{2\tau_0 \varrho}}{\left|{\frac{h_0(x)-h_0(x_0)-\langle Dh_0(x_0),x-x_0\rangle}{\tau_0 \varrho}}\right|}^2 {\,{{\rm{d}}}x}+c\,\rlap{-} \displaystyle {\int }_{B_{2\tau_0 \varrho}}{\left|{\frac{u_0-h_0}{\tau_0 \varrho}}\right|}^2 {\,{{\rm{d}}}x} \\ && \quad \stackrel{(3.11)}{ \leq} c(\tau_0 \varrho)^2\sup\limits_{B_{ \varrho/2}}\lvert { D^2 h_0}\rvert^2 +\frac{c\varepsilon}{\tau_0^{n+2}} \\ &&\quad \leq c\,\tau_0^{2}\rlap{-} \displaystyle {\int }_{B_{3 \varrho/4}}\lvert {Dh_0}\rvert^{2} {\,{{\rm{d}}}x} +\frac{c\varepsilon}{\tau_0^{n+2}}\\ &&\quad \stackrel{(3.10)}{ \leq} c\,\tau_0^{2}+\frac{c\varepsilon}{\tau_0^{n+2}}, \end{eqnarray} (3.12)

    where c\equiv c({\texttt{data}}) > 0 and where we have used the following property of \mathscr{A} -harmonic functions

    \begin{equation} \varrho^\gamma \sup\limits_{B_{ \varrho/2}}\lvert { D^2 h_0}\rvert^\gamma \leq c \, \rlap{-} \displaystyle {\int }_{B_{3 \varrho/4}}\lvert { D h_0}\rvert^\gamma {\,{{\rm{d}}}x}, \end{equation} (3.13)

    with \gamma > 1 and c depending on n , N , and on the ellipticity constants of \mathscr{A} .

    Now, choosing

    \varepsilon : = \tau_0^{n+2p},

    we have that this together with (3.9) gives that \varepsilon _0 \equiv \varepsilon _0({\texttt{data}}, \tau_0) and \varepsilon _1\equiv \varepsilon _1({\texttt{data}}, \tau_0) . Recalling the definition of u_0 in (3.7) and (3.12) we eventually arrive at

    \begin{eqnarray} && \rlap{-} \displaystyle {\int }_{B_{2\tau_0 \varrho}}\frac{\lvert {u-(u)_{ \varrho}-\langle( D u)_{ \varrho},x-x_0\rangle-\lvert {( D u)_ \varrho}\rvert^\frac{2-p}{2}\mathscr{E}(u;B_ \varrho)^{p/2}\left(h_0(x_0)-\langle D h_0(x_0),x-x_0\rangle\right)}\rvert^2}{(\tau_0 \varrho)^2}{\,{{\rm{d}}}x}\\ && \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \leq c\,\lvert {( D u)_ \varrho}\rvert^{2-p}\mathscr{E}(u;B_ \varrho)^p\tau_0^{2}, \end{eqnarray} (3.14)

    for c\equiv c({\texttt{data}}) > 0 . By a similar computation, always using (3.13), (3.10) and (3.11), we obtain that

    \begin{eqnarray*} d^{p-2}\rlap{-} \displaystyle {\int }_{B_{2\tau_0 \varrho}}\left|\frac{u_0-h_0(x_0)-\langle D h_0(x_0),x-x_0\rangle}{\tau_0 \varrho}\right|^{p} {\,{{\rm{d}}}x} \leq cd^{p-2}(\tau_0 \varrho)^{p}\sup\limits_{B_{ \varrho/2}}\lvert { D^{2}h_0}\rvert^{p}+\frac{c\,\varepsilon}{\tau_0^{n+p}} \leq c\,\tau_0^{p}. \end{eqnarray*}

    In this way, as for (3.14), by the definition of u_0 in (3.7), we eventually arrive at

    \begin{eqnarray} &&\rlap{-} \displaystyle {\int }_{B_{2\tau_0 \varrho}}\frac{\lvert {u-(u)_{ \varrho}-\langle( D u)_{ \varrho},x-x_0\rangle-\lvert {( D u)_ \varrho}\rvert^\frac{2-p}{2}\mathscr{E}(u;B_ \varrho)^{p/2}\left(h_0(x_0)-\langle D h_0(x_0),x-x_0\rangle\right)}\rvert^p}{(\tau_0 \varrho)^p}{\,{{\rm{d}}}x}\\ && \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \leq c \, d^{2-p}\lvert {( D u)_ \varrho}\rvert^\frac{p(2-p)}{2}\mathscr{E}(u;B_ \varrho)^\frac{p^2}{2} \tau_0^p\\ && \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \leq c \, \mathscr{E}(u;B_ \varrho)^p \tau_0^2, \end{eqnarray} (3.15)

    with c\equiv c({\texttt{data}}) .

    Denote now with \ell_{2\tau_0 \varrho} the unique affine function such that

    \ell_{2\tau_0 \varrho}\mapsto \min\limits_{\ell \ \text{affine}}\rlap{-} \displaystyle {\int }_{B_{2\tau_0 \varrho}}\lvert {u-\ell}\rvert^{2} {\,{{\rm{d}}}x}.

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

    \begin{equation} \rlap{-} \displaystyle {\int }_{B_{2\tau_0 \varrho}}\lvert {( D u)_ \varrho}\rvert^{p-2}{\left|{\frac{u-\ell_{2\tau_0 \varrho}}{2\tau_0 \varrho}}\right|}^2 + {\left|{\frac{u-\ell_{2\tau_0 \varrho}}{2\tau_0 \varrho}}\right|}^p {\,{{\rm{d}}}x} \leq c\, \tau^2 \mathscr{E}(u;B_ \varrho)^p. \end{equation} (3.16)

    Notice that we have also used the property that

    \rlap{-} \displaystyle {\int }_{B_ \varrho}\lvert {u-\ell_{ \varrho}}\rvert^p {\,{{\rm{d}}}x} \leq c\,\rlap{-} \displaystyle {\int }_{B_ \varrho}\lvert {u -\ell}\rvert^p{\,{{\rm{d}}}x},

    for p \geq 2 , c \equiv c(n, N, p) > 0 and for any affine function \ell ; see [33, Lemma 2.3].

    Recalling the definition of the excess functional \mathscr{E}(\cdot) , in (1.4), we can estimate the following quantity as follows

    \begin{eqnarray} \lvert {D\ell_{2\tau_0 \varrho}-(Du)_{ \varrho}}\rvert & \leq& \lvert {D\ell_{2\tau_0 \varrho}-(Du)_{2\tau_0 \varrho}}\rvert+\lvert {(Du)_{2\tau_0 \varrho}-(Du)_{ \varrho}}\rvert \\ & \leq& c\,\left(\rlap{-} \displaystyle {\int }_{B_{2\tau_0 \varrho}}\lvert {Du-(Du)_{2\tau_0 \varrho}}\rvert^{2} {\,{{\rm{d}}}x}\right)^{\frac{1}{2}}+\left(\rlap{-} \displaystyle {\int }_{B_{2\tau_0 \varrho}}\lvert {Du-(Du)_{ \varrho}}\rvert^{2} {\,{{\rm{d}}}x}\right)^{\frac{1}{2}} \\ &\stackrel{(2.1)}{ \leq}& \frac{c}{\tau_0^{n/2}}\left(\rlap{-} \displaystyle {\int }_{B_{ \varrho}}\lvert {Du-(Du)_{ \varrho}}\rvert^{2} {\,{{\rm{d}}}x}\right)^{\frac{1}{2}} \\ & = &\frac{c\lvert {(Du)_{ \varrho}}\rvert^{\frac{2-p}{2}}}{\tau_0^{n/2}}\left(\rlap{-} \displaystyle {\int }_{B_{ \varrho}}\lvert {(Du)_{ \varrho}}\rvert^{p-2}\lvert {Du-(Du)_{ \varrho}}\rvert^{2} {\,{{\rm{d}}}x}\right)^{\frac{1}{2}} \\ & \leq&\frac{c(n)}{\tau_0^{n/2}}\left(\frac{\mathscr{E}(u,B_ \varrho)}{\lvert {(Du)_ \varrho}\rvert}\right)^{\frac{p}{2}}\lvert {(Du)_{ \varrho}}\rvert, \end{eqnarray} (3.17)

    where we have used the following property of the affine function \ell_{2\tau_0 \varrho}

    \lvert { D \ell_{2\tau_0 \varrho} -( D u)_{2\tau_0 \varrho}}\rvert^p \leq c \, \rlap{-} \displaystyle {\int }_{B_{2\tau_0 \varrho}}\lvert { D u -( D u)_{2\tau_0 \varrho}}\rvert^p {\,{{\rm{d}}}x} ,

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

    Now, starting from (3.1) and (3.9), we further reduce the size of \varepsilon _0 such that

    \begin{equation} \left(\frac{\mathscr{E}(u,B_{ \varrho})}{\lvert {(Du)_{ \varrho}}\rvert}\right)^{\frac{p}{2}}\stackrel{(3.1)}{ \leq} \varepsilon _0^{\frac{p}{2}} \leq\frac{\tau_0^{n/2}}{8c(n)}, \end{equation} (3.18)

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

    \begin{eqnarray} \lvert {D\ell_{2\tau_0 \varrho}-(Du)_ \varrho}\rvert \leq\frac{\lvert {(Du)_ \varrho}\rvert}{8}. \end{eqnarray} (3.19)

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

    \begin{equation} \rlap{-} \displaystyle {\int }_{B_{2\tau_0 \varrho}}\lvert {D\ell_{2\tau_0 \varrho}}\rvert^{p-2}{\left|{\frac{u-\ell_{2\tau_0 \varrho}}{2\tau_0 \varrho}}\right|}^2 + {\left|{\frac{u-\ell_{2\tau_0 \varrho}}{2\tau_0 \varrho}}\right|}^p {\,{{\rm{d}}}x} \leq c\, \tau^2 \mathscr{E}(u;B_ \varrho)^p. \end{equation} (3.20)

    By triangular inequality and (3.19) we also get

    \lvert {D\ell_{2\tau_0 \varrho}}\rvert \geq \lvert {(Du)_{ \varrho}}\rvert-\lvert {D\ell_{2\tau_0 \varrho}-(Du)_{ \varrho}}\rvert \stackrel{(3.19)}{ \geq} \frac{7\lvert {(Du)_{ \varrho}}\rvert}{8}

    which, therefore, implies that

    \begin{eqnarray} && \rlap{-} \displaystyle {\int }_{B_{\tau_0 \varrho}}\lvert {D\ell_{2\tau_0 \varrho}}\rvert^{p-2}\lvert {Du-D\ell_{2\tau_0 \varrho}}\rvert^{2} {\,{{\rm{d}}}x}+\inf\limits_{z\in \mathbb{R}^{N\times n}}\rlap{-} \displaystyle {\int }_{B_{\tau_0 \varrho}}\lvert {Du-z}\rvert^p {\,{{\rm{d}}}x} \\ &&\qquad \stackrel{(2.7)}{ \leq} c\,\rlap{-} \displaystyle {\int }_{B_{2\tau_0 \varrho}}\lvert {D\ell_{2\tau_0 \varrho}}\rvert^{p-2}\left|\frac{u-\ell_{2\tau_0 \varrho}}{2\tau_0 \varrho}\right|^{2}+\left|\frac{u-\ell_{2\tau_0 \varrho}}{2\tau_0 \varrho}\right|^{p} {\,{{\rm{d}}}x} \\ &&\qquad \qquad +\frac{c}{\lvert {D\ell_{2\tau_0 \varrho}}\rvert^{p-2}}\left((2\tau_0 \varrho)^{m}\rlap{-} \displaystyle {\int }_{B_{2\tau_0 \varrho}}\lvert {f}\rvert^{m} {\,{{\rm{d}}}x}\right)^{\frac{2}{m}} \\ &&\qquad \stackrel{(3.20)}{ \leq} c\,\tau_0^{2}\mathscr{E}(u,B_ \varrho)^{p}+\frac{c\tau_0^{2-2n/m}}{\lvert {(Du)_ \varrho}\rvert^{p-2}}\left( \varrho^{m}\rlap{-} \displaystyle {\int }_{B_ \varrho}\lvert {f}\rvert^{m} {\,{{\rm{d}}}x}\right)^{\frac{2}{m}}, \end{eqnarray} (3.21)

    where c\equiv c({\texttt{data}}) > 0 . By triangular inequality, we can further estimate

    \begin{eqnarray*} &&\rlap{-} \displaystyle {\int }_{B_{\tau_0 \varrho}}\lvert {(Du)_{\tau_0 \varrho}}\rvert^{p-2}\lvert {Du-(Du)_{\tau_0 \varrho}}\rvert^{2} {\,{{\rm{d}}}x} \\ && \qquad \leq c\,\rlap{-} \displaystyle {\int }_{B_{\tau_0 \varrho}}\lvert {D\ell_{\tau_0 \varrho}-(Du)_{\tau_0 \varrho}}\rvert^{p-2}\lvert {Du-(Du)_{\tau_0 \varrho}}\rvert^{2} {\,{{\rm{d}}}x}\nonumber \\ &&\qquad\quad +c\,\rlap{-} \displaystyle {\int }_{B_{\tau_0 \varrho}}\lvert {D\ell_{2\tau_0 \varrho}-D\ell_{\tau_0 \varrho}}\rvert^{p-2}\lvert {Du-(Du)_{\tau_0 \varrho}}\rvert^{2} {\,{{\rm{d}}}x}\nonumber\\ &&\qquad\quad +c\,\rlap{-} \displaystyle {\int }_{B_{\tau_0 \varrho}}\lvert {D\ell_{2\tau_0 \varrho}}\rvert^{p-2}\lvert {Du-(Du)_{\tau_0 \varrho}}\rvert^{2} {\,{{\rm{d}}}x}\nonumber \\ &&\qquad = \mbox{I}_1+\mbox{I}_2+\mbox{I}_3, \end{eqnarray*}

    where c\equiv c(p) > 0 . We now separately estimate the previous integrals. We begin considering \mbox{I}_1 . By Young and triangular inequalities we get

    \begin{align*} \mbox{I}_1 \leq& c\lvert {D\ell_{\tau_0 \varrho}-(Du)_{\tau_0 \varrho}}\rvert^{p}+c\,\rlap{-} \displaystyle {\int }_{B_{\tau_0 \varrho}}\lvert {Du-(Du)_{\tau_0 \varrho}}\rvert^{p} {\,{{\rm{d}}}x}\nonumber \\ \leq& c\,\rlap{-} \displaystyle {\int }_{B_{\tau_0 \varrho}}\lvert {Du-(Du)_{\tau_0 \varrho}}\rvert^{p} {\,{{\rm{d}}}x}\\ \stackrel{(2.1)}{ \leq}& c\inf\limits_{z\in \mathbb{R}^N}\rlap{-} \displaystyle {\int }_{B_{\tau_0 \varrho}}\lvert {Du-z}\rvert^{p} {\,{{\rm{d}}}x}\nonumber \\ \stackrel{(3.21)}{ \leq}& c\,\tau_0^{2}\mathscr{E}(u,B_ \varrho)^{p}+\frac{c\tau_0^{2-2n/m}}{\lvert {(Du)_ \varrho}\rvert^{p-2}}\left( \varrho^{m}\rlap{-} \displaystyle {\int }_{B_ \varrho}\lvert {f}\rvert^{m} {\,{{\rm{d}}}x}\right)^{\frac{2}{m}}, \end{align*}

    with c\equiv c({\texttt{data}}) > 0 . In a similar fashion, we can treat the integral \mbox{I}_2

    \begin{align*} \mbox{I}_2 & \leq c\,\lvert {D\ell_{2\tau_0 \varrho}-D\ell_{\tau_0 \varrho}}\rvert^{p}+c\,\rlap{-} \displaystyle {\int }_{B_{\tau_0 \varrho}}\lvert {Du-(Du)_{\tau_0 \varrho}}\rvert^{p} {\,{{\rm{d}}}x}\nonumber \\ &\stackrel{(2.1)}{ \leq} c\,\rlap{-} \displaystyle {\int }_{B_{2\tau_0 \varrho}}\left|\frac{u-\ell_{2\tau_0 \varrho}}{2\tau_0 \varrho}\right|^{p} {\,{{\rm{d}}}x}+c\inf\limits_{z\in \mathbb{R}^{N\times n}}\rlap{-} \displaystyle {\int }_{B_{\tau_0 \varrho}}\lvert {Du-z}\rvert^{p} {\,{{\rm{d}}}x}\nonumber \\ &\stackrel{(3.20),(3.21)}{ \leq} c\,\tau_0^{2}\mathscr{E}(u,B_ \varrho)^{p}+\frac{c\tau_0^{2-2n/m}}{\lvert {(Du)_ \varrho}\rvert^{p-2}}\left( \varrho^{m}\rlap{-} \displaystyle {\int }_{B_ \varrho}\lvert {f}\rvert^{m} {\,{{\rm{d}}}x}\right)^{\frac{2}{m}}, \end{align*}

    where we have used the following property of the affine function \ell_{2\tau_0 \varrho}

    \lvert { D \ell_{2\tau_0 \varrho} - D \ell_{\tau_0 \varrho} }\rvert^p \leq c\,\rlap{-} \displaystyle {\int }_{B_{2\tau_0 \varrho}}\left|\frac{u-\ell_{2\tau_0 \varrho}}{2\tau_0 \varrho}\right|^{p} {\,{{\rm{d}}}x},

    for a given constant c \equiv c(n, p) > 0 ; see [33, Lemma 2.2]. Finally, the last integral \mbox{I}_3 can be treated recalling (3.21) and (2.1), i.e.,

    \mbox{I}_3 \leq c\,\tau_0^{2}\mathscr{E}(u,B_ \varrho)^{p}+\frac{c\tau_0^{2-2n/m}}{\lvert {(Du)_ \varrho}\rvert^{p-2}}\left( \varrho^{m}\rlap{-} \displaystyle {\int }_{B_ \varrho}\lvert {f}\rvert^{m} {\,{{\rm{d}}}x}\right)^{\frac{2}{m}}.

    All in all, combining the previous estimate

    \begin{eqnarray*} \mathscr{E}(u;B_{\tau_0 \varrho}) & \leq& c\,\tau_0^{2/p}\mathscr{E}(u,B_ \varrho)+\frac{c\tau_0^{2/p-2n/(mp)}}{\lvert {(Du)_ \varrho}\rvert^\frac{p-2}{p}}\left( \varrho^{m}\rlap{-} \displaystyle {\int }_{B_ \varrho}\lvert {f}\rvert^{m} {\,{{\rm{d}}}x}\right)^{\frac{2}{mp}}\\ &\stackrel{(3.2)}{ \leq}& c\,\tau_0^{2/p}\mathscr{E}(u,B_ \varrho)+c\tau_0^{2/p-2n/(mp)}\varepsilon _1^{2/p}\mathscr{E}(u;B_{\tau_0 \varrho})\\ & \leq& c_0\tau_0^{2/p}\mathscr{E}(u;B_{\tau_0 \varrho}), \end{eqnarray*}

    up to choosing \varepsilon _1 such that

    \varepsilon _1 \leq\tau_0^{n/m}.

    Step 2: proof of (3.5).

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

    \begin{eqnarray*} \mathscr{E}(u; B_{\tau_0 \varrho}(x_0))^\frac{p}{2} & \leq& \frac{2^{3p}}{\tau_0^{n/2}} \mathscr{E}(u; B_{ \varrho}(x_0))^\frac{p}{2}\\ &\stackrel{(3.4)}{ \leq}& \frac{2^{3p}}{\tau_0^{n/2}} \varepsilon _1^{-1} \lvert {( D u )_{B_ \varrho (x_0)}}\rvert^\frac{2-p}{2} \left( \varrho^m \rlap{-} \displaystyle {\int }_{B_{ \varrho}(x_{0})}\lvert {f}\rvert^m \ {\,{{\rm{d}}}x} \right)^\frac{1}{m} \\ &\stackrel{(3.1)}{ \leq}& \frac{2^{6(p-1)}}{\tau_0^{n(p-1)/p}} \varepsilon _0^\frac{p-2}{2} \varepsilon _1^{-1} \mathscr{E}(u;B_{\tau_0 \varrho})^\frac{2-p}{2} \left( \varrho^m \rlap{-} \displaystyle {\int }_{B_{ \varrho}(x_{0})}\lvert {f}\rvert^m \ {\,{{\rm{d}}}x} \right)^\frac{1}{m}. \end{eqnarray*}

    Multiplying both sides by \mathscr{E}(u; B_{\tau_0 \varrho})^\frac{p-2}{2} 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 z_0 = 0 .

    Lemma 3.1. Under assumptions (1.6) _{1, 2, 3} , (1.7) and (1.10), let u\in W^{1, p}(\varOmega, \mathbb{R}^{N}) be a local minimizer of functional (1.1). For every ball B_{ \varrho}(x_{0})\Subset \varOmega and any u_{0}\in \mathbb{R}^{N} , z_{0}\in \mathbb{R}^{N\times n} it holds that

    \begin{eqnarray} \mathscr{E}(u,z_0;B_{ \varrho/2}(x_0))^p & \leq& c \ \rlap{-} \displaystyle {\int }_{B_ \varrho(x_0)} \lvert {z_0}\rvert^{p-2}{\left|{\frac{u-\ell}{ \varrho}}\right|}^2 + {\left|{\frac{u-\ell}{ \varrho}}\right|}^p \ {\,{{\rm{d}}}x} \\ &&+\, c \left( \varrho^{m} \rlap{-} \displaystyle {\int }_{B_ \varrho(x_0)} \lvert {f}\rvert^m \ {\,{{\rm{d}}}x} \right)^\frac{p}{m(p-1)}, \end{eqnarray} (3.22)

    where \mathscr{E}(\cdot) is defined in (1.4), \ell(x): = u_{0}+\langle z_0, x-x_{0}\rangle and c\equiv c(n, N, \lambda, \varLambda, p) .

    Proof. The proof is analogous to estimate (2.7), up to treating in a different way the term \mbox{I}_2 in (2.9), taking in consideration the eventuality z_0 = 0 . Exploiting (1.10) and fact that \varphi_{1}\in W^{1, p}_{0}(B_{\tau_{2}}(x_{0}), \mathbb{R}^{N}) , an application of the Sobolev-Poincaré inequality yields

    \begin{eqnarray} \mbox{I}_{2}& \leq&\lvert {B_{\tau_{2}}(x_{0})}\rvert\left(\tau_{2}^{m}\rlap{-} \displaystyle {\int }_{B_{\tau_{2}}(x_{0})}\lvert {f}\rvert^{m}\ {\,{{\rm{d}}}x}\right)^{1/m}\left(\tau_{2}^{-m'}\rlap{-} \displaystyle {\int }_{B_{\tau_{2}}(x_{0})}\lvert {\varphi_{1}}\rvert^{m'}\ {\,{{\rm{d}}}x}\right)^{\frac{1}{m'}} \\ & \leq&\lvert {B_{\tau_{2}}(x_{0})}\rvert\left(\tau_{2}^{m}\rlap{-} \displaystyle {\int }_{B_{\tau_{2}}(x_{0})}\lvert {f}\rvert^{m}\ {\,{{\rm{d}}}x}\right)^{1/m}\left( \ \rlap{-} \displaystyle {\int }_{B_{\tau_{2}}(x_{0})}\left| \ \frac{\varphi_{1}}{\tau_{2}} \ \right|^{p^*}\ {\,{{\rm{d}}}x}\right)^{\frac{1}{p^*}} \\ & \leq&\lvert {B_{\tau_{2}}(x_{0})}\rvert\left(\tau_{2}^{m}\rlap{-} \displaystyle {\int }_{B_{\tau_{2}}(x_{0})}\lvert {f}\rvert^{m} {\,{{\rm{d}}}x}\right)^{1/m}\left( \ \rlap{-} \displaystyle {\int }_{B_{\tau_{2}}(x_{0})}\lvert {D\varphi_{1}}\rvert^{p} {\,{{\rm{d}}}x}\right)^{\frac{1}{p}} \\ & \leq&\varepsilon\int_{B_{\tau_{2}}(x_{0})}\lvert {V_{\lvert {z_{0}}\rvert}(D\varphi_{1})}\rvert^{2} {\,{{\rm{d}}}x}+\frac{c\lvert {B_{ \varrho}(x_{0})}\rvert}{\varepsilon ^{1/(p-1)}}\left( \varrho^{m}\rlap{-} \displaystyle {\int }_{B_{ \varrho}(x_{0})}\lvert {f}\rvert^{m} {\,{{\rm{d}}}x}\right)^{\frac{p}{m(p-1)}}, \end{eqnarray} (3.23)

    where c\equiv c(n, N, m) and we also used that \varrho/2 \leq\tau_{2} \leq \varrho . Hence, proceeding as in the proof of (2.7), we obtain that

    \begin{eqnarray*} && \int_{B_{\tau_{1}}(x_{0})}\lvert {V_{\lvert {z_{0}}\rvert}(Du-z_{0})}\rvert^{2} {\,{{\rm{d}}}x}\\ &&\quad \leq c\int_{B_{\tau_{2}}(x_{0})\setminus B_{\tau_{1}}(x_{0})}\lvert {V_{\lvert {z_{0}}\rvert}(Du-z_{0})}\rvert^{2}+\left| \ V_{\lvert {z_{0}}\rvert}\left(\frac{u-\ell}{\tau_{2}-\tau_{1}}\right)\ \right|^{2} {\,{{\rm{d}}}x} \nonumber \\ &&\qquad+\frac{c\lvert {B_{ \varrho}(x_{0})}\rvert}{\varepsilon ^{1/(p-1)}}\left( \varrho^{m}\rlap{-} \displaystyle {\int }_{B_{ \varrho}(x_{0})}\lvert {f}\rvert^{m} {\,{{\rm{d}}}x}\right)^{\frac{p}{m(p-1)}}, \end{eqnarray*}

    with c\equiv c(n, N, \lambda, \varLambda, 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 u\in W^{1, p}(\varOmega, \mathbb{R}^{N}) be a local minimizer of functional (1.1). For any B_ \varrho(x_{0}) \Subset \varOmega and any s \in (0, \infty) it holds that

    \begin{eqnarray} {\left|{ \ \rlap{-} \displaystyle {\int }_{B_ \varrho(x_{0})} \langle\lvert { D u}\rvert^{p-2} D u, D \varphi \rangle \ {\,{{\rm{d}}}x} }\right|} & \leq& s{\left\|{ D \varphi}\right\|}_{L^\infty(B_ \varrho (x_{0}))} \left( \ \rlap{-} \displaystyle {\int }_{B_ \varrho(x_{0})} \lvert { D u}\rvert^p \ {\,{{\rm{d}}}x} \right)^\frac{p-1}{p} \\ && +c\,\omega(s)^{-1}{\left\|{ D \varphi}\right\|}_{L^\infty(B_ \varrho (x_{0}))} \rlap{-} \displaystyle {\int }_{B_ \varrho (x_{0})} \lvert { D u}\rvert^p \ {\,{{\rm{d}}}x} \\ && +c\,{\left\|{ D \varphi}\right\|}_{L^\infty(B_ \varrho (x_{0}))} \left( \varrho^m \rlap{-} \displaystyle {\int }_{B_ \varrho(x_{0})} \lvert {f}\rvert^m \ {\,{{\rm{d}}}x}\right)^{1/m}, \end{eqnarray} (3.24)

    for any \varphi \in C^\infty_0(B_ \varrho(x_{0}), \mathbb{R}^N) , with c \equiv c(n, N, \varLambda, \lambda, 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])

    \begin{equation} \int_ \varOmega \big[\langle \partial F( D u), D \varphi \rangle - f \cdot \varphi \big] \ {\,{{\rm{d}}}x} = 0 \quad \mbox{for all } \varphi \in C^\infty_0( \varOmega, \mathbb{R}^N). \end{equation} (3.25)

    Now, fix \varphi \in C^\infty_0(B_ \varrho(x_{0}), \mathbb{R}^N) and split

    \begin{eqnarray*} &&{\left|{ \ \rlap{-} \displaystyle {\int }_{B_ \varrho(x_{0})} \langle\lvert { D u}\rvert^{p-2} D u, D \varphi \rangle \ {\,{{\rm{d}}}x} }\right|} \\ &&\qquad\stackrel{(3.25)}{ \leq} {\left|{ \ \rlap{-} \displaystyle {\int }_{B_ \varrho(x_{0})} \langle \partial F( D u) -\partial F(0)-\lvert { D u}\rvert^{p-2} D u, D \varphi \rangle \ {\,{{\rm{d}}}x} }\right|} + {\left|{ \ \rlap{-} \displaystyle {\int }_{B_ \varrho (x_0)} f \cdot \varphi \ {\,{{\rm{d}}}x}}\right|}\\ &&\qquad = : \mbox{I}_1 + \mbox{I}_2. \end{eqnarray*}

    We begin estimating the first integral \mbox{I}_1 . For s \in (0, \infty) we get

    \begin{eqnarray} \mbox{I}_1 & \leq& \frac{{\left\|{ D \varphi}\right\|}_{L^\infty(B_ \varrho (x_{0}))} }{\lvert {B_ \varrho (x_{0})}\rvert} \int_{B_ \varrho(x_{0}) \cap \{\lvert { D u}\rvert \leq\omega(s)\}} \lvert {\partial F( D u) -\partial F(0) -\lvert { D u}\rvert^{p-2} D u}\rvert \ {\,{{\rm{d}}}x} \\ && + \frac{{\left\|{ D \varphi}\right\|}_{L^\infty(B_ \varrho (x_{0}))} }{\lvert {B_ \varrho (x_{0})}\rvert} \int_{B_ \varrho(x_{0}) \cap \{\lvert { D u}\rvert > \omega(s)\}} \lvert {\partial F( D u) -\partial F(0) -\lvert { D u}\rvert^{p-2} D u}\rvert \ {\,{{\rm{d}}}x} \\ & \leq& s{\left\|{ D \varphi}\right\|}_{L^\infty(B_ \varrho (x_{0}))} \left( \ \rlap{-} \displaystyle {\int }_{B_ \varrho(x_{0})} \lvert { D u}\rvert^p\ {\,{{\rm{d}}}x} \right)^\frac{p-1}{p} \\ && + c\,\omega(s)^{-1}{\left\|{ D \varphi}\right\|}_{L^\infty(B_ \varrho (x_{0}))} \rlap{-} \displaystyle {\int }_{B_ \varrho(x_{0}) } \lvert { D u}\rvert^{p} \ {\,{{\rm{d}}}x} . \end{eqnarray} (3.26)

    On the other hand, the integral \mbox{I}_2 can be estimated as follows

    \begin{eqnarray*} \mbox{I}_2 & \leq& \left( \varrho^{m}\rlap{-} \displaystyle {\int }_{B_ \varrho(x_{0})}\lvert {f}\rvert^{m}\ {\,{{\rm{d}}}x}\right)^{1/m}\left( \ \rlap{-} \displaystyle {\int }_{B_ \varrho(x_{0})}{\left|{\frac{ \varphi}{ \varrho}}\right|}^{m'}\ {\,{{\rm{d}}}x}\right)^{\frac{1}{m'}}\nonumber \\ & \leq&\left( \varrho^{m}\rlap{-} \displaystyle {\int }_{B_{ \varrho}(x_{0})}\lvert {f}\rvert^{m}\ {\,{{\rm{d}}}x}\right)^{1/m}\left( \ \rlap{-} \displaystyle {\int }_{B_{ \varrho}(x_{0})}\left| \ \frac{ \varphi}{ \varrho} \ \right|^{p^*}\ {\,{{\rm{d}}}x}\right)^{\frac{1}{p^*}}\nonumber \\ & \leq&\left( \varrho^{m}\rlap{-} \displaystyle {\int }_{B_{ \varrho}(x_{0})}\lvert {f}\rvert^{m} {\,{{\rm{d}}}x}\right)^{1/m}\left( \ \rlap{-} \displaystyle {\int }_{B_{ \varrho}(x_{0})}\lvert { D \varphi}\rvert^{p} {\,{{\rm{d}}}x}\right)^{\frac{1}{p}}\nonumber \\ & \leq& {\left\|{ D \varphi}\right\|}_{L^\infty(B_ \varrho(x_0))}\left( \varrho^{m}\rlap{-} \displaystyle {\int }_{B_{ \varrho}(x_{0})}\lvert {f}\rvert^{m} {\,{{\rm{d}}}x}\right)^{1/m}. \end{eqnarray*}

    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 u\in W^{1, p}(\varOmega, \mathbb{R}^{N}) be a local minimizer of functional (1.1). Then, for any \chi \in (0, 1] and any \tau_1 \in (0, 2^{-10}) , there exists \varepsilon _2 \equiv \varepsilon _2(\mathit{{\texttt{data}}}, \chi, \tau_1) \in (0, 1) such that if the smallness conditions

    \begin{equation} \chi \lvert {( D u)_{B_ \varrho(x_0)}}\rvert \leq\mathscr{E}(u;B_ \varrho(x_0)), \quad \mathit{\mbox{and}} \quad \mathscr{E}(u;B_ \varrho(x_0)) \leq\varepsilon _2, \end{equation} (3.27)

    are satisfied on a ball B_ \varrho(x_0) \subset \mathbb{R}^n , then

    \begin{equation} \mathscr{E}(u;B_{\tau_1 \varrho}(x_0)) \leq c_1\tau_1^{\beta_1}\mathscr{E}(u;B_ \varrho(x_0)) +c_1 \left( \varrho^m \rlap{-} \displaystyle {\int }_{B_ \varrho(x_0)} \lvert {f}\rvert^m \ {\,{{\rm{d}}}x} \right)^\frac{1}{m(p-1)}, \end{equation} (3.28)

    for any \beta_1 \in (0, 2\alpha/p) , with \alpha \equiv \alpha (n, N, p) \in (0, 1) is the exponent in (3.34), and c_1 \equiv c_1(\mathit{{\texttt{data}}}, \chi) .

    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

    \begin{equation} \rlap{-} \displaystyle {\int }_{B_ \varrho}\lvert { D u}\rvert^p \ {\,{{\rm{d}}}x} \leq c_\chi \mathscr{E}(u;B_ \varrho)^p \qquad \text{with}\; c_\chi : = 2^p(1+\chi^{-p}). \end{equation} (3.29)

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

    \begin{eqnarray*} \rlap{-} \displaystyle {\int }_{B_ \varrho}\lvert { D u}\rvert^p \ {\,{{\rm{d}}}x} & \leq& 2^p\rlap{-} \displaystyle {\int }_{B_ \varrho}\lvert { D u-( D u)_{B_ \varrho}}\rvert^p \ {\,{{\rm{d}}}x} + 2^p\lvert {( D u)_{B_ \varrho}}\rvert^p\\ & \leq& 2^p \mathscr{E}(u;B_ \varrho)^p +\frac{2^p}{\chi^p} \mathscr{E}(u;B_ \varrho)^p. \end{eqnarray*}

    Consider now

    \kappa : = c_\chi \mathscr{E}(u;B_ \varrho) + \left( \big(\frac{ \varrho}{\varepsilon _3}\big)^m\rlap{-} \displaystyle {\int }_{B_ \varrho} \lvert {f}\rvert^m \, {\,{{\rm{d}}}x}\right)^\frac{1}{m(p-1)} \quad \mbox{and} \quad v_0 : = \frac{u}{ \kappa},

    for \varepsilon _3 \in (0, 1] , which will be fixed later on. Applying (3.24) to the function v_0 yields

    {\left|{ \ \rlap{-} \displaystyle {\int }_{B_{ \varrho/2}(x_{0})} \langle\lvert { D v_0}\rvert^{p-2} D v_0, D \varphi \rangle \, {\,{{\rm{d}}}x} }\right|} \stackrel{(3.27)_2,(3.29)}{ \leq} c {\left\|{ D \varphi}\right\|}_\infty \left(s +\omega(s)^{-1}\varepsilon _2 +\varepsilon _3 \right).

    For any \varepsilon > 0 and \vartheta \in (0, 1) and let \delta be the one given by [17, Lemma 1.1]. Then, up to choosing s , \varepsilon _2 and \varepsilon _3 sufficiently small, we arrive at

    c \left(s +\omega(s)^{-1}\varepsilon _2 +\varepsilon _3 \right) \leq\delta \| D \varphi\|_\infty^{p-1}.

    Then, Lemma 1.1 in [17] implies

    \left( \ \rlap{-} \displaystyle {\int }_{B_{ \varrho/2}} \lvert {V( D v_0)-V( D h)}\rvert^{2\vartheta} \, {\,{{\rm{d}}}x} \right)^\frac{1}{\vartheta} \leq c \varepsilon \ \rlap{-} \displaystyle {\int }_{B_{ \varrho/2}}\lvert { D u}\rvert^p \, {\,{{\rm{d}}}x} \stackrel{(3.29),(3.27)_2}{ \leq} c \varepsilon \varepsilon _2^p,

    up to taking \varepsilon as small as needed. Now, denoting with \mathfrak{h}_0: = h\kappa , we have that

    \left( \ \rlap{-} \displaystyle {\int }_{B_{ \varrho/2}} \lvert {V( D u)-V( D \mathfrak{h}_0)}\rvert^{2\vartheta} \, {\,{{\rm{d}}}x} \right)^\frac{1}{\vartheta} \leq \varepsilon \varepsilon _2^p \kappa^p.

    Now, we choose \vartheta: = (\mathfrak{s})'/2 , with \mathfrak{s} being the exponent given by (2.8). Note that by the proof of (2.8) it actually follows that \vartheta < 1 . Thus, choosing \varepsilon \varepsilon _2^p \kappa^p \leq\tau_1^{2n+4\alpha} (where \alpha \in (0, 1) is given by (3.34)) we arrive at

    \left( \ \rlap{-} \displaystyle {\int }_{B_{ \varrho/2}} \lvert {V( D u)-V( D \mathfrak{h}_0)}\rvert^{(\mathfrak{s})'} \, {\,{{\rm{d}}}x} \right)^\frac{1}{(\mathfrak{s})'} \leq c \, \tau_1^{n+2\alpha}.

    By Hölder's Inequality, we have that

    \begin{eqnarray} && \rlap{-} \displaystyle {\int }_{B_{ \varrho/2}} \lvert {V( D u)-V( D \mathfrak{h}_0)}\rvert^{2} {\,{{\rm{d}}}x} \\ &&\quad \leq\left( \ \rlap{-} \displaystyle {\int }_{B_{ \varrho/2} } \lvert {V( D u)-V( D \mathfrak{h}_0)}\rvert^{(\mathfrak{s})'} \, {\,{{\rm{d}}}x} \right)^\frac{1}{(\mathfrak{s})'}\left( \ \rlap{-} \displaystyle {\int }_{B_{ \varrho/2}} \lvert {V( D u)-V( D \mathfrak{h}_0)}\rvert^{\mathfrak{s}} \, {\,{{\rm{d}}}x} \right)^\frac{1}{\mathfrak{s}}. \end{eqnarray} (3.30)

    Hence, since by (2.3) V(z) \approx \lvert {z}\rvert^p , an application of estimates (2.8) and (3.29) now yields

    \begin{eqnarray} \left( \ \rlap{-} \displaystyle {\int }_{B_{ \varrho/2}} \lvert {V( D u)}\rvert^{\mathfrak{s}} \, {\,{{\rm{d}}}x} \right)^\frac{1}{\mathfrak{s}} & \leq& c\left( \ \rlap{-} \displaystyle {\int }_{B_{ \varrho/2}} \lvert { D u-( D u)_ \varrho}\rvert^{p_2} \, {\,{{\rm{d}}}x} \right)^\frac{p}{p_2} + c \lvert {( D u)_ \varrho}\rvert^p \\ & \leq& c \ \rlap{-} \displaystyle {\int }_{B_ \varrho} \lvert { D u}\rvert^p {\,{{\rm{d}}}x} + c \left( \varrho^m \rlap{-} \displaystyle {\int }_{B_ \varrho}\lvert {f}\rvert^m {\,{{\rm{d}}}x} \right)^\frac{p}{m(p-1)} +c\lvert {( D u )_ \varrho}\rvert^p\\ & \leq& c \, \mathscr{E}(u;B_ \varrho)^p + c \left( \varrho^m \rlap{-} \displaystyle {\int }_{B_ \varrho}\lvert {f}\rvert^m {\,{{\rm{d}}}x} \right)^\frac{p}{m(p-1)}, \end{eqnarray} (3.31)

    with c \equiv c({\texttt{data}}, \chi) .

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

    \begin{equation} \left( \ \rlap{-} \displaystyle {\int }_{B_{ \varrho/2}} \lvert {V( D \mathfrak{h}_0)}\rvert^{\mathfrak{s}} \, {\,{{\rm{d}}}x} \right)^\frac{1}{\mathfrak{s}} \leq c \ \rlap{-} \displaystyle {\int }_{B_ \varrho}\lvert { D \mathfrak{h}_0}\rvert^p {\,{{\rm{d}}}x} \leq c \ \rlap{-} \displaystyle {\int }_{B_ \varrho}\lvert { D u }\rvert^p {\,{{\rm{d}}}x} \leq c \, \mathscr{E}(u;B_ \varrho)^p. \end{equation} (3.32)

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

    \begin{eqnarray} \rlap{-} \displaystyle {\int }_{B_{ \varrho/2}} \lvert {V( D u)-V( D \mathfrak{h}_0)}\rvert^{2} {\,{{\rm{d}}}x} \leq c \, \tau_1^{n+2\alpha} \mathscr{E}(u;B_ \varrho)^p +c \, \tau_1^{n+2\alpha} \left( \varrho^m \rlap{-} \displaystyle {\int }_{B_ \varrho} \lvert {f}\rvert^m {\,{{\rm{d}}}x} \right)^\frac{p}{m(p-1)}. \end{eqnarray} (3.33)

    Let us recall that, for any \tau_1 \in (0, 2^{-10}) , given the p -harmonic function \mathfrak{h}_0 we have

    \begin{equation} \widetilde{\mathscr{E}}(\mathfrak{h}_0;B_{\tau_1 \varrho})^2 \leq c \tau_1^{2\alpha } \kappa^p, \qquad \alpha \equiv \alpha(n,N,p) \in (0,1). \end{equation} (3.34)

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

    \begin{eqnarray*} \lvert {( D u)_{\tau_1 \varrho}- ( D u)_ \varrho}\rvert & \leq& \left( \ \rlap{-} \displaystyle {\int }_{B_{\tau_1 \varrho}} \lvert { D u -( D u)_ \varrho}\rvert^p {\,{{\rm{d}}}x} \right)^\frac{1}{p}\\ & \leq& \tau_1^{-\frac{n}{p}} \left( \ \rlap{-} \displaystyle {\int }_{B_ \varrho} \lvert { D u -( D u)_ \varrho}\rvert^p {\,{{\rm{d}}}x} \right)^\frac{1}{p} \\ &\stackrel{(1.4),(3.27)_2}{ \leq}& \tau_1^{-\frac{n}{p}}\varepsilon _2. \end{eqnarray*}

    Thus, up to taking \varepsilon _2 sufficiently small, by the triangular inequality, we obtain that

    \frac{1}{2}\lvert {( D u)_{\tau_1 \varrho}}\rvert \leq\lvert {( D u)_ \varrho}\rvert \leq2 \lvert {( D u)_{\tau_1 \varrho}}\rvert.

    Hence, (2.2) yield

    \lvert {V_{\lvert {( D u)_{\tau_1 \varrho}}\rvert}(\cdot)}\rvert^2 \approx \lvert {V_{\lvert {( D u)_ \varrho}\rvert} (\cdot)}\rvert^2,

    and

    \lvert {V(( D u)_{\tau_1 \varrho})-V(( D u)_{ \varrho})}\rvert^2 \approx \lvert {V_{\lvert {( D u)_{ \varrho}}\rvert}\big(( D u)_{ \varrho}-( D u)_{\tau_1 \varrho}\big)}\rvert^2.

    Then,

    \begin{eqnarray*} \mathscr{E}(u;B_{\tau_1 \varrho})^p &\stackrel{(2.5)}{ \leq}& c\, \widetilde{\mathscr{E}}(u;B_{\tau_1 \varrho})^2\\ &\stackrel{(2.6)}{ \leq}& c \, \rlap{-} \displaystyle {\int }_{B_{\tau_1 \varrho}} \lvert {V( D u)-V(( D u)_{\tau_1 \varrho})}\rvert^2 {\,{{\rm{d}}}x}\\ & \leq& c \, \tau_1^{-n} \ \rlap{-} \displaystyle {\int }_{B_{ \varrho/2}} \lvert {V( D u)-V( D \mathfrak{h}_0)}\rvert^2 {\,{{\rm{d}}}x} \\ && + c \ \rlap{-} \displaystyle {\int }_{B_{\tau_1 \varrho}} \lvert {V( D \mathfrak{h}_0)-V(( D \mathfrak{h}_0)_{\tau_1 \varrho})}\rvert^2 {\,{{\rm{d}}}x}\\ &\stackrel{(2.6)}{ \leq}& c \, \tau_1^{-n} \ \rlap{-} \displaystyle {\int }_{B_{ \varrho/2}} \lvert {V( D u)-V( D \mathfrak{h}_0)}\rvert^2 {\,{{\rm{d}}}x} + c \ \widetilde{\mathscr{E}}(\mathfrak{h}_0,B_{\tau_1 \varrho}) \\ &\stackrel{(3.33),(3.34)}{ \leq}& c \, \tau_1^{2\alpha} \mathscr{E}(u;B_ \varrho)^p+ c \left( \varrho^m \rlap{-} \displaystyle {\int }_{B_ \varrho}\lvert {f}\rvert^m {\,{{\rm{d}}}x} \right)^\frac{p}{m(p-1)}, \end{eqnarray*}

    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 u\in W^{1, p}(\varOmega, \mathbb{R}^{N}) be a local minimizer of functional (1.1). Then, there exists \varepsilon _* \equiv \varepsilon _*(\mathit{{\texttt{data}}}) > 0 such that if the following condition

    \begin{equation} \mathscr{E}( D u; B_r) + \sup\limits_{ \varrho \leq r} \left( \varrho^m \rlap{-} \displaystyle {\int }_{B_ \varrho} \lvert {f}\rvert^m {\,{{\rm{d}}}x} \right)^\frac{1}{m(p-1)} < \varepsilon , \end{equation} (4.1)

    is satisfied on B_r\subset \varOmega , for some \varepsilon \in (0, \varepsilon _*] , then

    \begin{equation} \sup\limits_{ \varrho \leq r} \mathscr{E}( D u; B_ \varrho) < c_3\, \varepsilon , \end{equation} (4.2)

    for c_3 \equiv c_3(\mathit{{\texttt{data}}}) > 0 .

    Proof. For the sake of readability, since all balls considered in the proof are concentric to B_r(x_0) , we will omit denoting the center.

    Let us start fixing an exponent \beta \equiv \beta(\alpha, p) such that

    \begin{equation} 0 < \beta < \min\{\beta_0,\beta_1\} = :\beta_m, \end{equation} (4.3)

    where \beta_0 and \beta_1 are the exponents appearing in Propositions 3.1 and 3.2. Moreover, given the constant c_0 and c_1 from Propositions 3.1 and 3.2, choose \tau \equiv \tau({\texttt{data}}, \beta) such that

    \begin{equation} (c_0+c_1) \tau^{\beta_m -\beta} \leq\frac{1}{4}. \end{equation} (4.4)

    With the choice of \tau_0 as in (4.4) above, we can determine the constant \varepsilon _0 and \varepsilon _1 of Proposition 3.1. Now, we proceed applying Proposition 3.2 taking \chi \equiv \varepsilon _0 and \tau_1 as in (4.4) there. This determines the constant \varepsilon _2 and c_2 . We consider a ball B_r \subset \varOmega such that

    \begin{equation} \mathscr{E}( D u; B_r) < \varepsilon _2, \end{equation} (4.5)

    and

    \begin{equation} \sup\limits_{ \varrho \leq r} c_2 \left( \varrho^m \rlap{-} \displaystyle {\int }_{B_ \varrho} \lvert {f}\rvert^m {\,{{\rm{d}}}x}\right)^\frac{1}{m(p-1)} \leq\frac{\varepsilon _2}{4}, \end{equation} (4.6)

    where the constant c_2 : = c_1 +c_0 , with c_0 appearing in (3.5) and c_1 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 \varepsilon _0 \lvert {(D u)_{B_r}}\rvert \leq\mathscr{E}(D u; B_r) is satisfied then we obtain from (3.28), with \tau_1 \equiv \tau in (4.4) that

    \begin{eqnarray} \mathscr{E}(u;B_{\tau r}) & \leq& \frac{\tau^{\beta}}{4}\mathscr{E}(u;B_r) +c_2 \left(r^m \rlap{-} \displaystyle {\int }_{B_r} \lvert {f}\rvert^m \ {\,{{\rm{d}}}x} \right)^\frac{1}{m(p-1)} \\ & \leq& \frac{\tau^{\beta}}{4}\mathscr{E}(u;B_r) + \sup\limits_{ \varrho \leq r}c_2 \left( \varrho^m \rlap{-} \displaystyle {\int }_{B_ \varrho} \lvert {f}\rvert^m \ {\,{{\rm{d}}}x} \right)^\frac{1}{m(p-1)}\\ & \leq& \frac{\tau^{\beta}}{4}\mathscr{E}(u;B_r) + \frac{\varepsilon _2}{4} \leq\varepsilon _2, \end{eqnarray} (4.7)

    where the last inequality follows from (4.5) and (4.6). If on the other hand it holds \varepsilon _0 \lvert {(D u)_{B_r}}\rvert \geq \mathscr{E}(D u; B_r) , 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

    \mathscr{E}( D u ; B_{\tau^j r}) < \varepsilon _2 \quad \mbox{for any } j \geq 0,

    and the estimate

    \mathscr{E}(u;B_{\tau^{j+1} r}) \leq \frac{\tau^{\beta}}{4}\mathscr{E}(u;B_{\tau^j r}) +c_2 \left((\tau^j r)^m \rlap{-} \displaystyle {\int }_{B_{\tau^j r}} \lvert {f}\rvert^m \ {\,{{\rm{d}}}x} \right)^\frac{1}{m(p-1)},

    holds true. By the inequality above we have that for any k \geq 0

    \begin{eqnarray*} \mathscr{E}(u;B_{\tau^{k+1} r}) & \leq& \frac{\tau^{\beta(k +1)}}{4}\mathscr{E}(u;B_{r}) +c_2 \sum\limits_{j = 0}^{k} (\tau^\beta)^{j-k} \left((\tau^j r)^m \rlap{-} \displaystyle {\int }_{B_{\tau^j r}} \lvert {f}\rvert^m \ {\,{{\rm{d}}}x} \right)^\frac{1}{m(p-1)} \\ & \leq& \tau^{\beta(k +1)}\mathscr{E}(u;B_{r}) +c_2 \sup\limits_{ \varrho \leq r} \left( \varrho^m \rlap{-} \displaystyle {\int }_{B_rr} \lvert {f}\rvert^m {\,{{\rm{d}}}x}\right)^\frac{1}{m(p-1)}. \end{eqnarray*}

    Applying a standard interpolation argument we conclude that, for any t \leq r , it holds

    \begin{equation} \mathscr{E}( D u, B_s) \leq c_3 \left(\frac{s}{r}\right)^\beta\mathscr{E}( D u, B_r) +c_3 \sup\limits_{ \varrho \leq r} \left( \varrho^m \rlap{-} \displaystyle {\int }_{B_rr} \lvert {f}\rvert^m {\,{{\rm{d}}}x}\right)^\frac{1}{m(p-1)}, \end{equation} (4.8)

    where c_3 \equiv c_3({\texttt{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 1 \leq m < n and any \mathscr{O} \subset \varOmega , with positive measure, we have that

    \begin{equation} \|f\|_{L^{m}(\mathscr{O})} \leq\left(\frac{n}{n-m}\right)^{1/m}\lvert {\mathscr{O}}\rvert^{1/m-1/n}\|f\|_{L^{n,\infty}(\mathscr{O})}. \end{equation} (4.9)

    Indeed, fix \bar{\lambda} which will be chosen later on. Then, we have that

    \begin{equation} \|f\|_{L^m(\mathscr{O})}^m = m \int_0^{\bar{\lambda}}\lambda^m \lvert {\{x \in \mathscr{O}:\lvert {f}\rvert > \lambda\}}\rvert\frac{{\rm d}\lambda}{\lambda} +m \int_{\bar{\lambda}}^\infty\lambda^m \lvert {\{x \in \mathscr{O}:\lvert {f}\rvert > \lambda\}}\rvert\frac{{\rm d}\lambda}{\lambda}. \end{equation} (4.10)

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

    \int_0^{\bar{\lambda}}\lambda^m\lvert {\{x \in \mathscr{O}:\lvert {f}\rvert > \lambda\}}\rvert\frac{{\rm d}\lambda}{\lambda} \leq\frac{\bar{\lambda}^m\lvert {\mathscr{O} }\rvert }{m}.

    On the other hand, the second integral can be estimated recalling the definition of the L^{n, \infty}(\mathscr{O}) -norm. Indeed,

    \int_{\bar{\lambda}}^\infty\lambda^m \lvert {\{x \in \mathscr{O}:\lvert {f}\rvert > \lambda\}}\rvert\frac{{\rm d}\lambda}{\lambda} \leq\|f\|_{L^{n,\infty}(\mathscr{O})}^n \int_{\bar{\lambda}}^\infty \frac{{\rm d}\lambda}{\lambda^{1+n-m}} \leq\frac{\|f\|_{L^{n,\infty}(\mathscr{O})}^n}{(n-m)\bar{\lambda}^{n-m}}.

    Hence, putting all the estimates above in (4.10), choosing \bar{\lambda}: = \|f\|_{L^{n, \infty}(\mathscr{O})}/\lvert {\mathscr{O}}\rvert^{1/n} , we obtain (4.9).

    Now, recalling condition (1.2) we have that

    \begin{eqnarray*} \left( \varrho^m \rlap{-} \displaystyle {\int }_{B_ \varrho}\lvert {f}\rvert^m \, {\,{{\rm{d}}}x}\right)^{1/m} & \leq& \left(\frac{n}{n-m}\right)^{1/m}\lvert {B_1}\rvert^{-1/n}\|f\|_{L^{n,\infty}( \varOmega)}\notag\\ &\stackrel{(1.10)}{ \leq}& \left(\frac{4^{n/m}}{\lvert {B_1}\rvert}\right)^{1/n}\|f\|_{L^{n,\infty}( \varOmega)} \stackrel{(1.2)}{ \leq} \varepsilon _*, \end{eqnarray*}

    where \varepsilon _* is the one obtained in the proof of Proposition 4.1. From this it follows that, we can choose a radius \varrho_1 such that

    \begin{equation} \sup\limits_{ \varrho \leq \varrho_1}c_2\left( \varrho^m \rlap{-} \displaystyle {\int }_{B_ \varrho(x)}\lvert {f}\rvert^m \, {\,{{\rm{d}}}x}\right)^{1/m(p-1)} \leq\frac{\varepsilon _*}{4c_3}. \end{equation} (4.11)

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

    \varOmega_u: = \left\{x_0 \in \varOmega: \, \exists B_ \varrho(x_0)\Subset \varOmega \, \text{with}\; \varrho \leq \varrho_1 \, : \mathscr{E}( D u, B_ \varrho(x_0)) < \varepsilon _*/(4c_3)\right\},

    thus fixing \varrho_{x_0}: = \varrho_1 and \varepsilon _{x_0}: = \varepsilon _*/(4c_3) . We first star noting that the the set \varOmega_u defined in (1.4) is such that \lvert { \varOmega setminus \varOmega_u}\rvert = 0 . Indeed, let us consider the set

    \begin{equation} \mathscr{L}_u: = \left\{x_0 \in \varOmega: \liminf\limits_{ \varrho \to 0} \widetilde{\mathscr{E}}(u;B_ \varrho(x_0))^2 = 0\right\}, \end{equation} (4.12)

    which is such that \lvert { \varOmega setminus \mathscr{L}_u}\rvert = 0 by standard Lebesgue's Theory. Moreover, by (2.5) it follows that

    \mathscr{L}_u: = \left\{x_0 \in \varOmega: \liminf\limits_{ \varrho \to 0} \mathscr{E}(u;B_ \varrho(x_0)) = 0\right\},

    so that, \mathscr{L}_u \subset \varOmega_u and we eventually obtained that \lvert { \varOmega setminus \varOmega_u}\rvert = 0 . Now we show that \varOmega_u is open. Let us fix x_0 \in \varOmega_u and find a radius \varrho_{x_0} \leq \varrho_{1} such that

    \begin{equation} \mathscr{E}( D u, B_{ \varrho_{x_0}}(x_0)) < \frac{\varepsilon _*}{4c_3}. \end{equation} (4.13)

    By absolute continuity of the functional \mathscr{E}(\cdot) we have that there exists an open neighbourhood \mathscr{O}(x_0) such that, for any x \in \mathscr{O}(x_0) it holds

    \begin{equation} \mathscr{E}( D u, B_{ \varrho_{x_0}}(x)) < \frac{\varepsilon _*}{4c_3} \quad \text{and}\; B_{ \varrho_{x_0}}(x) \Subset \varOmega. \end{equation} (4.14)

    This prove that \varOmega_u is open. Now let us start noting that (4.11) and (4.14) yield that condition (4.1) is satisfied with B_r \equiv B_{ \varrho_{x_0}}(x) . Hence, an application of Proposition 4.1 yields

    \sup\limits_{t \leq \varrho_{x_0}}\mathscr{E}( D u, B_t(x)) < \varepsilon _*,

    for any x \in \mathscr{O}(x_0) . 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.



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