1 Definitions and preliminaries

Let Ω be an open bounded set in \(\mathbb{R}^{N}\). We define the family of kernels \(( k_{\delta } ) _{\delta >0}\) as a set of radial positive functions fulfilling the following properties:

  1. (1)
    $$ \frac{1}{C_{N}} \int _{B ( 0,\delta ) }k_{\delta } \bigl( \vert s \vert \bigr) \,ds=1, $$

    where

    $$ C_{N}=\frac{1}{\operatorname{meas} ( S^{N-1} ) } \int _{S^{N-1}} \vert \sigma \cdot \mathbf{e} \vert ^{p}\,d\mathcal{H}^{N-1} ( \sigma ), $$

    \(\mathcal{H}^{N-1}\) stands for the \(( N-1 ) \)-dimensional Hausdorff measure on the unit sphere \(S^{N-1}\), e is any unit vector in \(\mathbb{R}^{N}\), \(p>1\), and \(B(x,\delta )\) is the ball with center x and radius δ.

  2. (2)

    \(\operatorname{supp}k_{\delta }\subset B ( 0,\delta ) \).

We define the nonlocal operator \(\mathcal{B}_{h}\) in \(L^{p} ( \Omega ) \times L^{p} ( \Omega ) \) by

$$ \mathcal{B}_{h} ( u,u ) = \int _{\Omega } \int _{\Omega }H \bigl( x^{\prime },x \bigr) \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u \bigl( x^{\prime } \bigr) -u ( x ) \bigr\vert ^{p} \,dx^{\prime }\,dx, $$

where \(H ( x^{\prime },x ) = \frac{h ( x^{\prime } ) +h ( x ) }{2}\), \(h\in \mathcal{H}\),

$$ \mathcal{H}\doteq \bigl\{ h:\Omega \rightarrow \mathbb{R}\mid h ( x ) \in {}[ h_{\min },h_{\max }]\text{ a.e. }x\in \Omega, h=0\text{ in } \mathbb{R}^{N}\setminus \Omega \bigr\} , $$

and \(0< h_{\min }< h_{\max }\) are given constants.

For \(h=1\), the following compactness result is well known (see, e.g., [4] and [9, proof of Theorem 1.2, p. 12]).

Theorem 1

Let \(( u_{\delta } ) _{\delta }\) be a sequence uniformly bounded in \(L^{p} ( \Omega ) \), and let C be a positive constant such that

$$ \int _{\Omega } \int _{\Omega } \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u_{\delta } \bigl( x^{\prime } \bigr) -u_{\delta } ( x ) \bigr\vert ^{p} \,dx^{\prime }\,dx\leq C $$
(1.1)

for any δ. Then from \(( u_{\delta } ) _{\delta }\) we can extract a subsequence, still denoted by \(( u_{\delta } ) _{\delta }\), and we can find \(u\in W^{1,p} ( \Omega ) \) such that \(u_{\delta }\rightarrow u\) strongly in \(L^{p} ( \Omega ) \) as \(\delta \rightarrow 0\) and

$$ \lim_{\delta \rightarrow 0} \int _{\Omega } \int _{\Omega } \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u_{\delta } \bigl( x^{\prime } \bigr) -u_{\delta } ( x ) \bigr\vert ^{p} \,dx^{\prime }\,dx\geq \int _{\Omega } \bigl\vert \nabla u ( x ) \bigr\vert ^{p}\,dx. $$
(1.2)

Even though several authors are involved in the proof, we refer to estimate (1.2) as Ponce’s inequality.

1.1 The objective

Our goal is to prove the following extension of (1.2):

$$ \lim_{\delta \rightarrow 0} \int _{\Omega } \int _{\Omega }H \bigl( x^{ \prime },x \bigr) \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u_{\delta } \bigl( x^{\prime } \bigr) -u_{\delta } ( x ) \bigr\vert ^{p}\,dx^{\prime }\,dx\geq \int _{\Omega }h ( x ) \bigl\vert \nabla u ( x ) \bigr\vert ^{p}\,dx, $$
(1.3)

where Ω is an open bounded set, \(H ( x^{\prime },x ) = \frac{h ( x^{\prime })+h(x ) }{2}\), and \(h\in \mathcal{H}\).

As we will see, inequality (1.3) is equivalent to (1.2) for measurable sets, that is,

$$ \lim_{\delta \rightarrow 0} \int _{E} \int _{E} \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u_{\delta } \bigl( x^{\prime } \bigr) -u_{\delta } ( x ) \bigr\vert ^{p} \,dx^{\prime }\,dx\geq \int _{E} \bigl\vert \nabla u ( x ) \bigr\vert ^{p}\,dx $$
(1.4)

for all measurable sets E in Ω.

1.2 Motivation and organization of the paper

The context in which we locate the present paper is the study of the nonlocal p-Laplacian problem. Before proceeding, we make precise some notation. We define the spaces

$$ L_{0}^{p} ( \Omega _{\delta } ) = \bigl\{ u\in L^{p} ( \Omega _{\delta } ):u=0\text{ in }\mathbb{R}^{N} \setminus \Omega \bigr\} $$

and

$$ X= \bigl\{ u\in L_{0}^{p} ( \Omega _{\delta } ): \mathcal{B} ( u,u ) < \infty \bigr\} , $$

where

$$ \Omega _{\delta }=\Omega \cup \biggl( \bigcup_{x\in \partial \Omega }B ( x,\delta ) \biggr), $$

\(\mathcal{B}=\mathcal{B}_{1}\), and \(\mathcal{B}_{h}\) is the operator defined in \(X\times X\) by

$$ \mathcal{B}_{h} ( u,v ) = \int _{\Omega _{\delta }} \int _{ \Omega _{\delta }}H \bigl( x^{\prime },x \bigr) \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u \bigl( x^{\prime } \bigr) -u ( x ) \bigr\vert ^{p-2} \bigl( u \bigl( x^{\prime } \bigr) -u ( x ) \bigr) \bigl( v \bigl( x^{\prime } \bigr) -v ( x ) \bigr) \,dx^{\prime }\,dx. $$

We define now the following nonlocal variational problem: given \(f\in L^{p^{\prime }} ( \Omega ) \), where \(p^{\prime }=\frac{p}{p-1}\) and \(p>1\), find \(u\in X\) such that

$$ \mathcal{B}_{h} ( u,w ) = ( f,w ) _{L^{p^{ \prime }} ( \Omega ) \times L^{p} ( \Omega ) } \quad\text{in }X. $$
(1.5)

Note that (1.5) is equivalent to

$$\begin{aligned} &\int _{\Omega _{\delta }} \int _{\Omega _{\delta }}H \bigl( x^{\prime },x \bigr) k_{\delta } \bigl( \bigl\vert x^{\prime }-x \bigr\vert \bigr) \frac{ \vert u ( x^{\prime } ) -u ( x ) \vert ^{p-2} ( u ( x^{\prime } ) -u ( x ) ) ( w ( x^{\prime } ) -w ( x ) ) }{ \vert x^{\prime }-x \vert ^{p}} \,dx^{\prime }\,dx \\ &\quad= \int _{\Omega _{\delta }}fw\,dx \end{aligned}$$
(1.6)

for all w∈ X. Since the existence and uniqueness of solution for this problem is a well-known fact, for h fixed and any δ, there exists a solution \(u_{\delta }\). The aim is to check whether the sequence of solutions \(( u_{\delta } ) _{\delta }\) converges to the solution of the corresponding local p-Laplacian equation. This convergence (or G-convergence) clearly entails the study of the minimization principle

$$ \min_{w\in X} \biggl\{ \frac{1}{p}\mathcal{B}_{h} ( w,w ) - \int _{\Omega }f ( x ) w ( x ) \,dx \biggr\} , $$

and, consequently, this task inevitably leads us to the study of the problem posed above; [1–3, 5] are some references where this type of convergence is analyzed.

The paper is organized by means of three sections containing different proofs of (1.3) and (1.4).

2 First proof

Our essential tool in to generalize (1.3) is a convenient Vitali covering of the set Ω (see [11, Chap. 4, Sect. 3, p. 109.] for details or [6, Chap. 2, Sect. 2, p. 26] for an elegant proof in the case of Lebesgue-measurable sets). Recall that the family \(\{ V_{i} \} _{i\in I}\) is a Vitali covering for \(\Omega \subset \mathbb{R}^{N}\) if with any \(x\in \Omega \) we can associate a number \(\alpha >0\), a sequence of \(V_{i}\), and a sequence of balls \(B ( x,\epsilon _{i} ) \) such that \(V_{i}\subset B ( x,\epsilon _{i} ) \) and \(\vert V_{i} \vert \geq \alpha \vert B ( x, \epsilon _{i} ) \vert \), where \(\epsilon _{i}\rightarrow 0\) as \(i\rightarrow \infty \).

Theorem 2

(Vitali covering theorem)

Let \(\mathcal{A}= \{ V_{i} \} _{k\in K}\) be a Vitali covering of closed subsets of \(\mathbb{R}^{N}\) for Ω. There is a sequence of \(( i_{j} ) _{j}\in K\) such that \(\vert \Omega \setminus \bigcup_{j}V_{i_{j}} \vert =0\) and the sets \(( V_{i_{j}} ) _{j}\) are pairwise disjoint.

A particular and useful version of this chief result is the following:

Proposition 1

Let \(\Omega \subset \mathbb{R}^{N}\) be an open bounded set, let K be a compact set included in Ω, and let ξ be a nonnegative function in \(L^{1} ( \Omega \times \Omega ) \). Then there is a sequence of pairwise disjoint closed balls \(( \overline{B}_{i} ) \subset \Omega \) such that \(\vert K\setminus \bigcup_{i=1}^{\infty }\overline{B}_{i} \vert =0\) and

$$ \iint _{K\times K}\xi \bigl( x^{\prime },x \bigr) \,dx^{\prime }\,dx\geq \sum_{i=1}^{\infty } \iint _{\overline{B}_{i}\times \overline{B}_{i}} \xi \bigl( x^{\prime },x \bigr) \,dx^{\prime }\,dx. $$

Proof

Since K is a compact inside Ω and Ω is open, we have \(d\doteq \operatorname{dist} ( K,\mathbb{R}^{N}\setminus \Omega ) >0\). In particular, any closed ball \(\overline{B}_{i}=\overline{B ( x,r ) }\subset \Omega \) for any \(r< d\). Moreover, the family \(\mathcal{F}= \{ \overline{B ( x,s ) }:x\in K, s< r/2 \} \) is a Vitali covering of K, because every point of K is contained in an arbitrarily small ball belonging to \(\mathcal{F}\). Consequently, there are disjoint balls \(\overline{B}_{i}\) such that \(\vert K\setminus \bigcup_{i=1}^{\infty }\overline{B}_{i} \vert =0\). This covering also serves to approximate \(K\times K\) because \(\vert ( K\times K ) \setminus ( \bigcup_{i,j=1}^{ \infty } ( \overline{B}_{i}\times \overline{B}_{j} ) ) \vert =0\), and therefore

$$ \iint _{K\times K}\xi \bigl( x^{\prime },x \bigr) \,dx^{\prime }\,dx= \sum_{i,j=1}^{\infty } \iint _{\overline{B}_{i}\times \overline{B}_{j}} \xi \bigl( x^{\prime },x \bigr) \,dx^{\prime }\,dx\geq \sum_{i=1}^{ \infty } \iint _{\overline{B}_{i}\times \overline{B}_{i}}\xi \bigl( x^{ \prime },x \bigr) \,dx^{\prime }\,dx. $$

 □

In a first step, we assume that h is continuous a.e. in Ω. We adapt [7, Lemma 7.9, p. 129] to prove our key result.

Proposition 2

Let \(\Omega \subset \mathbb{R}^{N}\) be an open bounded set such that \(\vert \partial \Omega \vert =0\), and let f be a positive a.e. continuous function on Ω. Let \(r_{k}:\Omega \setminus N\rightarrow \mathbb{R}^{+}\) be a sequence of functions, where N is the set of discontinuity points of f. There exist a set of points \(\{ a_{ki} \} _{i}\subset \Omega \setminus N\) and positive numbers \(\{ \epsilon _{ki} \} _{i} \) such that for each k, \(\epsilon _{ki}\leq r_{k} ( a_{ki} ) \),

$$\begin{aligned} & \{ a_{ki}+\epsilon _{ki}\overline{\Omega } \} \quad\textit{are pairwise disjoint}, \\ &\overline{\Omega } =\bigcup_{i} \{ a_{ki}+\epsilon _{ki} \overline{\Omega } \} \cup N_{k},\quad\textit{where } \vert N_{k} \vert =0, \end{aligned}$$

and

$$ \int _{\Omega }f ( x ) \xi ( x ) \,dx=\sum _{i}f ( a_{ki} ) \int _{a_{ki}+\epsilon _{ki}\Omega }\xi ( x ) \,dx+o ( 1 ) \quad\textit{as }k\rightarrow + \infty $$
(2.1)

for all \(\xi \in L^{1} ( \Omega ) \).

Proof

Let \(C=\Omega \setminus N\) be the set of points of continuity of f. We define the families

$$ \mathcal{F}_{k}= \biggl\{ a+\epsilon \overline{\Omega }\subset \Omega:a \in C, \epsilon \leq r_{k} ( a ), \bigl\vert f ( x ) -f ( a ) \bigr\vert \leq \frac{1}{k} \text{ for any }x\in a+\epsilon \Omega \biggr\} . $$

For each fixed \(k>0\), the family \(\mathcal{F}_{k}\) covers C (and Ω) in the sense of Vitali. Thus, Theorem 2 allows us to choose a numerable sequence of disjoints sets \(\{ a_{kj}+\epsilon _{kj}\overline{\Omega } \} _{j}\in \mathcal{F}_{k}\) such that \(\vert \overline{\Omega }\setminus \bigcup_{j} \{ a_{kj}+ \epsilon _{kj}\overline{\Omega } \} \vert =0\). Since f is continuous at \(a_{kj}\), the sequence \(\epsilon _{kj}\) can be chosen so that

$$ \bigl\vert f ( x ) -f ( a_{kj} ) \bigr\vert \leq \frac{1}{k}\quad\text{for any }x\in a_{kj}+\epsilon _{kj} \Omega \text{ and any }j. $$

Consequently,

$$\begin{aligned} & \biggl\vert \int _{\Omega }\xi ( x ) f ( x ) \,dx- \sum _{j}f ( a_{kj} ) \int _{a_{kj}+\epsilon _{kj}\Omega } \xi ( x ) \,dx \biggr\vert \\ &\quad = \biggl\vert \sum_{j} \int _{a_{kj}+\epsilon _{kj}\Omega } \bigl( f ( x ) -f ( a_{kj} ) \bigr) \xi ( x ) \,dx \biggr\vert \\ &\quad \leq \sum_{j} \int _{a_{kj}+\epsilon _{kj}\Omega } \bigl\vert \bigl( f ( x ) -f ( a_{kj} ) \bigr) \bigr\vert \bigl\vert \xi ( x ) \bigr\vert \,dx \\ &\quad \leq \frac{1}{k}\sum_{j} \int _{a_{kj}+\epsilon _{kj}\Omega } \bigl\vert \xi ( x ) \bigr\vert \,dx \\ &\quad =\frac{1}{k} \Vert \xi \Vert _{L^{1} ( \Omega ) }. \end{aligned}$$

 □

2.1 Application

We apply the previous analysis to the integral

$$ I= \int _{\Omega } \int _{\Omega }H \bigl( x^{\prime },x \bigr) \xi _{ \delta } \bigl( x^{\prime },x \bigr) \,dx^{\prime }\,dx, $$

where

$$ \xi _{\delta } \bigl( x^{\prime },x \bigr) = \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u_{\delta } \bigl( x^{\prime } \bigr) -u_{\delta } ( x ) \bigr\vert . $$
(2.2)

We consider \(\Omega \times \Omega \) instead of Ω, and now \(f ( x^{\prime },x ) \) is the symmetric function \(H ( x^{\prime },x ) = \frac{h ( x^{\prime } ) +h ( x ) }{2}\) with \(h\in \mathcal{H}\). We assume that h is continuous, and we take \(\bigcup_{i,j} ( a_{ki}+\epsilon _{ki}\Omega ) \times ( a_{kj}+\epsilon _{kj}\Omega ) \), the union of a family of pairwise of disjoint sets covering \(\Omega \times \Omega \). Then, according to the previous discussion, we trivially deduce

$$\begin{aligned} I & =\sum_{i,j}H ( a_{ki},a_{kj} ) \int _{a_{ki}+ \epsilon _{ki}\Omega } \int _{a_{kj}+\epsilon _{kj}\Omega } \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u_{\delta } \bigl( x^{\prime } \bigr) -u_{\delta } ( x ) \bigr\vert ^{p} \,dx^{\prime }\,dx+o ( 1 ) \\ & \geq \sum_{i}H ( a_{ki},a_{ki} ) \int _{a_{ki}+ \epsilon _{ki}\Omega } \int _{a_{ki}+\epsilon _{ki}\Omega } \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u_{\delta } \bigl( x^{\prime } \bigr) -u_{\delta } ( x ) \bigr\vert ^{p} \,dx^{\prime }\,dx+o ( 1 ) \\ & =\sum_{i}h ( a_{ki} ) \int _{a_{ki}+\epsilon _{ki} \Omega }\int _{a_{ki}+\epsilon _{ki}\Omega } \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u_{\delta } \bigl( x^{\prime } \bigr) -u_{\delta } ( x ) \bigr\vert ^{p} \,dx^{\prime }\,dx+o ( 1 ). \end{aligned}$$

We pass to the limit as \(\delta \rightarrow 0\) in I: we use (1.1), Fatou’s lemma and (1.2) for open sets to derive

$$\begin{aligned} \liminf_{\delta \rightarrow 0} I \geq{}& \liminf_{\delta \rightarrow 0} \sum_{i}h ( a_{ki} ) \int _{a_{ki}+ \epsilon _{ki}\Omega } \int _{a_{ki}+\epsilon _{ki}\Omega } \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u_{\delta } \bigl( x^{\prime } \bigr) -u_{\delta } ( x ) \bigr\vert ^{p} \,dx^{\prime }\,dx+o ( 1 ) \\ \geq{}& \sum_{i}h ( a_{ki} ) \biggl( \liminf_{\delta \rightarrow 0} \int _{a_{ki}+\epsilon _{ki}\Omega } \int _{a_{ki}+ \epsilon _{ki}\Omega } \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u_{\delta } \bigl( x^{\prime } \bigr) -u_{\delta } ( x ) \bigr\vert ^{p} \,dx^{ \prime }\,dx \biggr)\\ &{} +o ( 1 ) \\ \geq{}& \sum_{i}h ( a_{ki} ) \biggl( \int _{a_{ki}+ \epsilon _{ki}\Omega } \bigl\vert \nabla u ( x ) \bigr\vert ^{p}\,dx \biggr) +o ( 1 ). \end{aligned}$$

If we take limits as \(k\rightarrow +\infty \), then this estimate gives

$$ \liminf_{\delta \rightarrow 0} I\geq \lim_{k\rightarrow + \infty }\sum _{i}h ( a_{ki} ) \int _{a_{ki}+\epsilon _{ki} \Omega } \bigl\vert \nabla u ( x ) \bigr\vert ^{p}\,dx. $$

By using again Proposition 2 the last inequality clearly provides inequality (1.3).

Remark 1

The analysis and conclusion we have just arrived at remain valid if we consider any open set \(O\subset \Omega \) such that \(\vert \partial O \vert =0\). We can go a step further: we have

$$ \liminf_{\delta \rightarrow 0} \int _{O} \int _{O}F \bigl( x^{\prime },x \bigr) \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u_{\delta } \bigl( x^{\prime } \bigr) -u_{\delta } ( x ) \bigr\vert ^{p}\,dx^{ \prime }\,dx\geq \int _{O}F ( x,x ) \bigl\vert \nabla u ( x ) \bigr\vert ^{p}\,dx $$
(2.3)

for any symmetric nonnegative continuous function \(F\in L^{\infty } ( O\times O ) \).

2.2 Extension to the case of measurable functions

Let now h be just measurable; without loss of generality, \(\operatorname{supp}H\subset \Omega \times \Omega \) and \(H=0\) otherwise. By Luzin’s theorem (see [10, Theorem 2.24, p. 62]), given arbitrary \(\epsilon >0\), there exists a continuous function \(G\in C_{c} ( \Omega \times \Omega ) \) such that \(\sup G ( x,y ) \leq \sup H ( x,y ) \) and \(G ( x,y ) =H ( x,y ) \) for any \(( x,y ) \in ( \Omega \times \Omega ) \setminus \mathcal{E}\), where \(\mathcal{E}\) is a measurable set such that \(\vert \mathcal{E} \vert <\epsilon ^{2}\). Since H is symmetric, we can assume that \(( \Omega \times \Omega ) \setminus \mathcal{E=} ( \Omega \setminus E ) \times ( \Omega \setminus E ) \), where \(E\subset \Omega \) is a measurable set such that \(\vert E \vert <\epsilon \).

At this stage, we consider any compact set \(K\subset \Omega \setminus E\subset \Omega \). Since Ω is open, we can use Proposition 1: there is a number \(r>0\) such that the family \(\mathcal{F}= \{ \overline{B ( x,s ) }:x\in K, s< r/2 \} \) is a Vitali covering of K, and therefore there exists a sequence of pairwise disjoint closed balls \(( \overline{B}_{i} ) _{i=1}^{\infty }\subset \mathcal{F}\) such that \(\vert K\setminus \bigcup_{i=1}^{\infty }\overline{B}_{i} \vert \), \(\overline{B}_{i}\subset \Omega \), and

$$\begin{aligned} & \int _{\Omega } \int _{\Omega }H \bigl( x^{\prime },x \bigr) \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u_{\delta } \bigl( x^{\prime } \bigr) -u_{\delta } ( x ) \bigr\vert ^{p}\,dx^{\prime }\,dx \\ &\quad \geq \int _{\Omega \setminus E} \int _{\Omega \setminus E}H \bigl( x^{ \prime },x \bigr) \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u_{\delta } \bigl( x^{\prime } \bigr) -u_{\delta } ( x ) \bigr\vert ^{p}\,dx^{\prime }\,dx \\ &\quad \geq \iint _{ ( \Omega \setminus E ) \times ( \Omega \setminus E ) }G \bigl( x^{\prime },x \bigr) \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u_{\delta } \bigl( x^{\prime } \bigr) -u_{\delta } ( x ) \bigr\vert ^{p}\,dx^{\prime }\,dx \\ &\quad \geq \iint _{K\times K}G \bigl( x^{\prime },x \bigr) \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u_{\delta } \bigl( x^{\prime } \bigr) -u_{\delta } ( x ) \bigr\vert ^{p}\,dx^{\prime }\,dx \\ &\quad \geq \sum_{i} \iint _{\overline{B}_{i}\times \overline{B}_{i}}G \bigl( x^{\prime },x \bigr) \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u_{\delta } \bigl( x^{\prime } \bigr) -u_{\delta } ( x ) \bigr\vert ^{p}\,dx^{\prime }\,dx. \end{aligned}$$

We take the limits as \(\delta \rightarrow 0\) to get

$$\begin{aligned} & \liminf_{\delta \rightarrow 0} \int _{\Omega } \int _{\Omega }H \bigl( x^{ \prime },x \bigr) \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u_{\delta } \bigl( x^{\prime } \bigr) -u_{\delta } ( x ) \bigr\vert ^{p}\,dx^{\prime }\,dx \\ & \quad\geq \liminf_{\delta \rightarrow 0}\sum_{i} \iint _{\overline{B}_{i}\times \overline{B}_{i}}G \bigl( x^{\prime },x \bigr) \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u_{\delta } \bigl( x^{\prime } \bigr) -u_{\delta } ( x ) \bigr\vert ^{p}\,dx^{\prime }\,dx \\ & \quad\geq \sum_{i} \int _{B_{i}}G ( x,x ) \bigl\vert \nabla u ( x ) \bigr\vert ^{p}\,dx \\ &\quad = \int _{K}G ( x,x ) \bigl\vert \nabla u ( x ) \bigr\vert ^{p}\,dx, \end{aligned}$$

where the second inequality is true because of (2.3) and Fatou’s lemma. Then, since K is any compact set in \(\Omega \setminus E\), we obtain

$$\begin{aligned} & \liminf_{\delta \rightarrow 0} \int _{\Omega } \int _{\Omega }H \bigl( x^{ \prime },x \bigr) \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u_{\delta } \bigl( x^{\prime } \bigr) -u_{\delta } ( x ) \bigr\vert ^{p}\,dx^{\prime }\,dx \\ &\quad \geq \int _{\Omega \setminus E}G ( x,x ) \bigl\vert \nabla u ( x ) \bigr\vert ^{p}\,dx \\ & \quad= \int _{\Omega \setminus E}H ( x,x ) \bigl\vert \nabla u ( x ) \bigr\vert ^{p}\,dx \\ &\quad = \int _{\Omega }h ( x ) \bigl\vert \nabla u ( x ) \bigr\vert ^{p}\,dx- \int _{E}h ( x ) \bigl\vert \nabla u ( x ) \bigr\vert ^{p}\,dx. \end{aligned}$$

By letting \(\epsilon \downarrow 0\) and using \(\vert E \vert \leq \epsilon \), we obtain (1.3), that is,

$$ \liminf_{\delta \rightarrow 0} \int _{\Omega } \int _{\Omega }H \bigl( x^{ \prime },x \bigr) \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u_{\delta } \bigl( x^{\prime } \bigr) -u_{\delta } ( x ) \bigr\vert ^{p}\,dx^{\prime }\,dx\geq \int _{\Omega }H ( x,x ) \bigl\vert \nabla u ( x ) \bigr\vert ^{p}\,dx. $$
(2.4)

Finally, to circumvent the assumption \(\vert \partial \Omega \vert =0\), the procedure we follow is identical to that just employed. Take any compact set K included in Ω. Since Ω is assumed to be open, thanks to Proposition 1, K can be exhaustively covered by the union of a numerable sequence of pairwise disjoint closed balls \(\overline{B}_{i}\in \mathcal{F}= \{ \overline{B ( x,s ) }:x\in K, s< r/2 \} \subset \Omega \), \(i=1,2,\dots \). Then we realize that

$$\begin{aligned} &\int _{\Omega } \int _{\Omega }H \bigl( x^{\prime },x \bigr) \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u_{\delta } \bigl( x^{\prime } \bigr) -u_{\delta } ( x ) \bigr\vert ^{p}\,dx^{\prime }\,dx \\ &\quad\geq \sum_{i} \int _{B_{i}} \int _{B_{i}}H \bigl( x^{\prime },x \bigr) \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u_{\delta } \bigl( x^{\prime } \bigr) -u_{\delta } ( x ) \bigr\vert ^{p}\,dx^{\prime }\,dx. \end{aligned}$$
(2.5)

By taking into account that \(\vert \partial B_{i} \vert =0\) we can apply (2.3) and Fatou’s lemma in (2.5) to obtain

$$ \liminf_{\delta \rightarrow 0} \int _{\Omega } \int _{\Omega }H \bigl( x^{ \prime },x \bigr) \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u_{\delta } \bigl( x^{\prime } \bigr) -u_{\delta } ( x ) \bigr\vert ^{p}\,dx^{\prime }\,dx\geq \int _{K}H ( x,x ) \bigl\vert \nabla u ( x ) \bigr\vert ^{p}\,dx. $$

Since \(K\subset \Omega \) is arbitrary, we arrive at (2.4) for any open set Ω,

$$\begin{aligned} &\liminf_{\delta \rightarrow 0} \int _{\Omega } \int _{\Omega }H \bigl( x^{ \prime },x \bigr) \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u_{\delta } \bigl( x^{\prime } \bigr) -u_{\delta } ( x ) \bigr\vert ^{p}\,dx^{\prime }\,dx \\ &\quad\geq \int _{\Omega }H ( x,x ) \bigl\vert \nabla u ( x ) \bigr\vert ^{p}\,dx. \end{aligned}$$
(2.6)

2.3 Corollary

We apply (2.4) to the case \(F ( x^{\prime },x ) =I_{G\times G} ( x^{\prime },x ) \), where G is any measurable set included in Ω: on the one hand, (2.6) guarantees

$$\begin{aligned} &\liminf_{\delta \rightarrow 0} \int _{\Omega } \int _{\Omega }F \bigl( x^{\prime },x \bigr) \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u_{\delta } \bigl( x^{\prime } \bigr) -u_{\delta } ( x ) \bigr\vert ^{p}\,dx^{\prime }\,dx\\ &\quad\geq \int _{\Omega }F ( x,x ) \bigl\vert \nabla u ( x ) \bigr\vert ^{p}\,dx \\ &\quad= \int _{G}I_{G} ( x ) \bigl\vert \nabla u ( x ) \bigr\vert ^{p}\,dx= \int _{G} \bigl\vert \nabla u ( x ) \bigr\vert ^{p}\,dx, \end{aligned}$$

and, on the other hand, it is obvious that

$$\begin{aligned} &\int _{\Omega } \int _{\Omega }F \bigl( x^{\prime },x \bigr) \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u_{\delta } \bigl( x^{\prime } \bigr) -u_{\delta } ( x ) \bigr\vert ^{p}\,dx^{\prime }\,dx\\ &\quad= \int _{G} \int _{G} \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u_{\delta } \bigl( x^{\prime } \bigr) -u_{\delta } ( x ) \bigr\vert ^{p} \,dx^{ \prime }\,dx. \end{aligned}$$

Consequently, (1.4) is proved for any measurable set \(G\subset \Omega \).

3 A second proof

We firstly prove (1.4) and then (1.3). By having a look at the work done in the previous section we will be able to provide a straightforward proof of (1.4). Indeed, if E is a measurable set included in Ω, then we can find a compact set \(K\subset E\) such that \(\vert E\setminus K \vert \) is arbitrarily small. Proposition 1 ensures the existence of a numerable sequence of pairwise disjoint balls \(\overline{B}_{i}\in \mathcal{F}\) such that \(\vert K\setminus \bigcup_{i=1}^{\infty }\overline{B}_{i} \vert =0\), \(\overline{B}_{i}\subset \Omega \) for any i and

$$\begin{aligned} & \int _{E} \int _{E} \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u_{\delta } \bigl( x^{\prime } \bigr) -u_{\delta } ( x ) \bigr\vert ^{p} \,dx^{\prime }\,dx \\ & \quad\geq \int _{K} \int _{K} \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u_{\delta } \bigl( x^{\prime } \bigr) -u_{\delta } ( x ) \bigr\vert ^{p} \,dx^{\prime }\,dx \\ & \quad\geq \sum_{i} \int _{B_{i}} \int _{B_{i}} \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u_{\delta } \bigl( x^{\prime } \bigr) -u_{\delta } ( x ) \bigr\vert ^{p} \,dx^{\prime }\,dx. \end{aligned}$$

We apply (1.2) for open sets and Fatou’s lemma in the last chain of inequalities to derive

$$ \liminf_{\delta \rightarrow 0} \int _{E} \int _{E} \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u_{\delta } \bigl( x^{\prime } \bigr) -u_{\delta } ( x ) \bigr\vert ^{p} \,dx^{\prime }\,dx\geq \sum_{i} \int _{B_{i}} \bigl\vert \nabla u ( x ) \bigr\vert ^{p}\,dx= \int _{K} \bigl\vert \nabla u ( x ) \bigr\vert ^{p}\,dx. $$

Since \(K\subset E\) is arbitrary, we arrive at (1.4), that is,

$$ \liminf_{\delta \rightarrow 0} \int _{E} \int _{E} \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u_{\delta } \bigl( x^{\prime } \bigr) -u_{\delta } ( x ) \bigr\vert ^{p} \,dx^{\prime }\,dx\geq \int _{E} \bigl\vert \nabla u ( x ) \bigr\vert ^{p}\,dx. $$
(3.1)

3.1 Corollary

We prove (1.3). Let h be a given simple function defined in Ω. Then h can be written as \(h ( x ) =\sum_{i=1}^{m}h_{i}I_{B_{i}} ( x ) \), where \(\{ B_{i} \} \) is a finite covering of disjoint measurable subsets of Ω, and \(( h_{i} ) _{i}\) is a set of numbers such that \(h_{\min }\leq h_{i}\leq h_{\max }\). Consequently, we can easily check that

$$\begin{aligned} I & \doteq \int _{\Omega } \int _{\Omega }H \bigl( x^{\prime },x \bigr) k_{ \delta } \bigl( \bigl\vert x^{\prime }-x \bigr\vert \bigr) \frac{ \vert u_{\delta } ( x^{\prime } ) -u_{\delta } ( x ) \vert ^{p}}{ \vert x^{\prime }-x \vert ^{p}} \,dx^{ \prime }\,dx \\ & \geq \sum_{i=1}^{m}h_{i} \int _{B_{i}} \int _{B_{i}}k_{\delta } \bigl( \bigl\vert x^{\prime }-x \bigr\vert \bigr) \frac{ \vert u_{\delta } ( x^{\prime } ) -u_{\delta } ( x ) \vert ^{p}}{ \vert x^{\prime }-x \vert ^{p}}\,dx^{ \prime }\,dx. \end{aligned}$$

Using inequality (1.4) for measurable sets that we have just proved, we straightforwardly infer

$$ \liminf_{\delta \rightarrow 0} I\geq \sum_{i=1}^{m}h_{i} \int _{B_{i}} \bigl\vert \nabla u ( x ) \bigr\vert ^{p}\,dx= \int _{ \Omega }h ( x ) \bigl\vert \nabla u ( x ) \bigr\vert ^{p}\,dx. $$

Let h be a measurable function. By recalling that any measurable function h can be pointwise approximated by an increasing sequence \(( s_{n} ) _{n}\) of simple functions we can write

$$\begin{aligned} & \liminf_{\delta \rightarrow 0} \int _{\Omega } \int _{\Omega }H \bigl( x^{\prime },x \bigr) k_{\delta } \bigl( \bigl\vert x^{\prime }-x \bigr\vert \bigr) \frac{ \vert u_{\delta } ( x^{\prime } ) -u_{\delta } ( x ) \vert ^{p}}{ \vert x^{\prime }-x \vert ^{p}} \,dx^{\prime }\,dx \\ &\quad =\liminf_{\delta \rightarrow 0} \int _{\Omega }h ( x ) \int _{\Omega }k_{\delta } \bigl( \bigl\vert x^{\prime }-x \bigr\vert \bigr) \frac{ \vert u_{\delta } ( x^{\prime } ) -u_{\delta } ( x ) \vert ^{p}}{ \vert x^{\prime }-x \vert ^{p}}\,dx^{\prime }\,dx \\ &\quad \geq \liminf_{\delta \rightarrow 0} \int _{\Omega }s_{n} ( x ) \int _{\Omega _{\delta }}k_{\delta } \bigl( \bigl\vert x^{\prime }-x \bigr\vert \bigr) \frac{ \vert u_{\delta } ( x^{\prime } ) -u_{\delta } ( x ) \vert ^{p}}{ \vert x^{\prime }-x \vert ^{p}}\,dx^{\prime }\,dx \\ &\quad \geq \int _{\Omega }s_{n} ( x ) \bigl\vert \nabla u ( x ) \bigr\vert ^{p}\,dx. \end{aligned}$$

It suffices to take the limits in n and apply the monotone convergence theorem to establish (1.3).

4 A third proof

The idea is reproducing the arguments from [9]. In a first step, we assume that \(h:\overline{\Omega }\rightarrow [ h_{\min },h_{\max } ] \) is a continuous function. Moreover, without loss of generality, we suppose that h is a continuous function in the set \(\Omega _{s}=\Omega \cup \{ \bigcup_{p\in \partial \Omega }B ( p,s ) \} \), where s is a fixed positive number.

Now, for the proof of (1.3), the key idea is extending the Stein inequality (see [8, Lemma 4, p. 245]) in the following sense: by using Jensen’s inequality and performing a change of variables we deduce the inequality

$$\begin{aligned} & \int _{\Omega } \int _{\Omega }H_{r} \bigl( x^{\prime },x \bigr) k_{ \delta } \bigl( \bigl\vert x^{\prime }-x \bigr\vert \bigr) \frac{ \vert u_{\delta } ( x^{\prime } ) -u_{\delta } ( x ) \vert ^{p}}{ \vert x^{\prime }-x \vert ^{p}}\,dx^{ \prime }\,dx \\ &\quad \geq \int _{\Omega _{-r}} \int _{\Omega _{-r}}H \bigl( x^{\prime },x \bigr) k_{\delta } \bigl( \bigl\vert x^{\prime }-x \bigr\vert \bigr) \frac{ \vert u_{r,\delta } ( x^{\prime } ) -u_{r,\delta } ( x ) \vert ^{p}}{ \vert x^{\prime }-x \vert ^{p}} \,dx^{ \prime }\,dx \end{aligned}$$

for any \(\delta < r\), where \(u_{r,\delta }=\eta _{r}\ast u_{\delta }\), \(\eta _{r} ( x ) =\frac{1}{r^{N}}\eta ( \frac{x}{r} ) \), \(x\in \mathbb{R}^{N}\), η is a nonnegative radial function from \(C_{c}^{\infty } ( B ( 0,1 ) ) \) such that \(\int \eta ( x ) \,dx=1\),

$$ H_{r} \bigl( x^{\prime },x \bigr) = \frac{ ( \eta _{r}\ast h ) ( x^{\prime } ) + ( \eta _{r}\ast h ) ( x ) }{2},$$

and \(\Omega _{-r}= \{ x\in \Omega:\operatorname{dist}(x,\partial \Omega )>r \} \). By the continuity of H in \(\Omega _{s}\times \Omega _{s}\) we know that \(H_{r} ( x^{\prime },x ) \rightarrow H ( x^{\prime },x ) \) uniformly on compact sets of \(\Omega _{s}\times \Omega _{s}\), whereby, for any \(\epsilon >0\), we can choose \(r_{0}>0\) such that

$$ \biggl\vert \int _{\Omega } \int _{\Omega } \bigl( H \bigl( x^{\prime },x \bigr) -H_{r} \bigl( x^{\prime },x \bigr) \bigr) k_{\delta } \bigl( \bigl\vert x^{\prime }-x \bigr\vert \bigr) \frac{ \vert u_{\delta } ( x^{\prime } ) -u_{\delta } ( x ) \vert ^{p}}{ \vert x^{\prime }-x \vert ^{p}} \,dx^{ \prime }\,dx \biggr\vert \leq \epsilon C $$

for any \(r< r_{0}\) and uniformly in \(\delta >0\). Then

$$\begin{aligned} & \lim_{\delta \rightarrow 0} \int _{\Omega } \int _{\Omega }H \bigl( x^{ \prime },x \bigr) k_{\delta } \bigl( \bigl\vert x^{\prime }-x \bigr\vert \bigr) \frac{ \vert u_{\delta } ( x^{\prime } ) -u_{\delta } ( x ) \vert ^{p}}{ \vert x^{\prime }-x \vert ^{p}}\,dx^{\prime }\,dx \\ & \quad\geq \lim_{\delta \rightarrow 0} \int _{\Omega _{-r}} \int _{\Omega _{-r}}H \bigl( x^{\prime },x \bigr) k_{\delta } \bigl( \bigl\vert x^{\prime }-x \bigr\vert \bigr) \frac{ \vert u_{r,\delta } ( x^{\prime } ) -u_{r,\delta } ( x ) \vert ^{p}}{ \vert x^{\prime }-x \vert ^{p}} \,dx^{\prime }\,dx-\epsilon C \end{aligned}$$

for any \(r< r_{0}\). At this point, we notice that Proposition 1 from [8, p. 242] can be modified by including the term \(H ( x^{\prime },x ) \) within the integrand; this is factually what Remark 1 establishes. Then passing to the limit as \(\delta \rightarrow 0\) and using the convergence of \(\rho _{r}\ast u_{\delta }\rightarrow \rho _{r}\ast u\) in \(C^{2} ( \overline{\Omega }_{-r} ) \), we get

$$\begin{aligned} &\lim_{\delta \rightarrow 0} \int _{\Omega _{-r}} \int _{\Omega _{-r}}H \bigl( x^{\prime },x \bigr) k_{\delta } \bigl( \bigl\vert x^{\prime }-x \bigr\vert \bigr) \frac{ \vert u_{r,\delta } ( x^{\prime } ) -u_{r,\delta } ( x ) \vert ^{p}}{ \vert x^{\prime }-x \vert ^{p}} \,dx^{\prime }\,dx\\ &\quad\geq \int _{\Omega _{-r}}h ( x ) \bigl\vert \nabla ( \rho _{r} \ast u ) ( x ) \bigr\vert ^{p}\,dx^{\prime }\,dx. \end{aligned}$$

Consequently, letting \(r\rightarrow 0\) in this inequality and taking into account that \(\nabla ( \rho _{r}\ast u ) \) strongly converges to ∇u in \(L^{p} ( \Omega ) \), we derive

$$ \lim_{\delta \rightarrow 0} \int _{\Omega } \int _{\Omega }H \bigl( x^{ \prime },x \bigr) k_{\delta } \bigl( \bigl\vert x^{\prime }-x \bigr\vert \bigr) \frac{ \vert u_{\delta } ( x^{\prime } ) -u_{\delta } ( x ) \vert ^{p}}{ \vert x^{\prime }-x \vert ^{p}}\,dx^{\prime }\,dx\geq \int _{\Omega }h ( x ) \bigl\vert \nabla u ( x ) \bigr\vert ^{p}\,dx^{\prime }\,dx-\epsilon C. $$

Now, since ϵ is arbitrarily small, the statement is proved under the assumption that h is continuous in \(\Omega _{s}\).

If \(h:\Omega \rightarrow [ h_{\min },h_{\max } ] \) is a measurable function, then we extend it by zero to \(\Omega _{s}\) and then apply Luzin’s theorem to this extended function. The remaining details follow along the lines of Sect. 2.2.