Neuroimmunomodulation by estrogen in health and disease

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

1.   Introduction

In this paper we provide a limiting partial regularity criterion for vector-valued minimizers $u : \varOmega\subset \mathbb{R}^{n} \to \mathbb{R}^N$, $n\ge 2$, $N > 1$, of nonhomogeneous, quasiconvex variational integrals as:

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

and the related borderline function space criterion

This is the content of our main theorem.

Theorem 1.1. 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 a number $\varepsilon _*\equiv \varepsilon _*({{\texttt{data}}}) > 0$ such that if

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

Moreover, the set $\varOmega_u$ can be characterized as follows

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

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

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 $L(n, \infty)$.

1.1.   Outline of the paper

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.

1.2.   Structural assumptions

In (1.1), the integrand $F\colon \mathbb{R}^{N\times n}\to \mathbb{R}$ satisfies

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

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.,

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

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

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

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

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

see [24, Chapter 5].

2.   Preliminaries

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

2.1.   Notation

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

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

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 $t_{*}: = \max\left\{nt/(n+t), 1\right\}$.

2.2.   Tools for nonlinear problems

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

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

and

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

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.,

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

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

2.3.   Basic regularity results

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

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

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

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

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

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

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:

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

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

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

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

where we set

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

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

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

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

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. □

3.   Excess decay estimate

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.

3.1.   The nondegenerate scenario

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

and

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

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

is satisfied on $B_ \varrho(x_0)$, then

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

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

and (3.6) follows.

Consider now

and

Let us note that we have

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

Then,

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

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

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.,

such that

and

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

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

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

Now, choosing

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

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

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

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

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

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

Notice that we have also used the property that

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

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

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

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

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

By triangular inequality and (3.19) we also get

which, therefore, implies that

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

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

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

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

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.,

All in all, combining the previous estimate

up to choosing $\varepsilon _1$ such that

Step 2: proof of (3.5).

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

Multiplying both sides by $\mathscr{E}(u; B_{\tau_0 \varrho})^\frac{p-2}{2}$ we get the desired estimate. □

3.2.   The degenerate scenario

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

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

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

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

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])

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

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

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

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

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

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

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

Consider now

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

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

Then, Lemma 1.1 in ^[17] implies

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

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

By Hölder's Inequality, we have that

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

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

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

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

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

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

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

Hence, (2.2) yield

and

Then,

and the desired estimate (3.28) follows. □

4.   Proof of the main result

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

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

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

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

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

and

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

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

and the estimate

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

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

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

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

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

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

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

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

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

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

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

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

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

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

for any $x \in \mathscr{O}(x_0)$. Thus concluding the proof. □

Use of AI tools declaration

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

Acknowledgments

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.

Conflict of interest

The author declares no conflict of interest.