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# Advances in Nonlinear Analysis

Editor-in-Chief: Radulescu, Vicentiu / Squassina, Marco

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Volume 8, Issue 1

# Absence of Lavrentiev gap for non-autonomous functionals with (p,q)-growth

Antonio Esposito
/ Francesco Leonetti
/ Pier Vincenzo Petricca
Published Online: 2016-12-20 | DOI: https://doi.org/10.1515/anona-2016-0198

## Abstract

We consider non-autonomous functionals of the form $\mathcal{ℱ}\left(u,\mathrm{\Omega }\right)={\int }_{\mathrm{\Omega }}f\left(x,Du\left(x\right)\right)𝑑x$, where $u:\mathrm{\Omega }\to {ℝ}^{N}$, $\mathrm{\Omega }\subset {ℝ}^{n}$. We assume that $f\left(x,z\right)$ grows at least as ${|z|}^{p}$ and at most as ${|z|}^{q}$. Moreover, $f\left(x,z\right)$ is Hölder continuous with respect to x and convex with respect to z. In this setting, we give a sufficient condition on the density $f\left(x,z\right)$ that ensures the absence of a Lavrentiev gap.

MSC 2010: 49N60

## 1 Introduction

We consider variational integrals of the form

$\mathcal{ℱ}\left(u,\mathrm{\Omega }\right)={\int }_{\mathrm{\Omega }}f\left(x,Du\left(x\right)\right)𝑑x$(1.1)

where $u:\mathrm{\Omega }\subset {ℝ}^{n}\to {ℝ}^{N}$, $n\ge 2$, $N\ge 1$, $f:\mathrm{\Omega }×{ℝ}^{nN}\to ℝ$ is a Caratheodory function, and Ω is bounded and open. Moreover, we assume that, for exponents $1 and constants $\nu ,L\in \left(0,+\mathrm{\infty }\right)$, $c\in \left[0,+\mathrm{\infty }\right)$, we have

$\nu {|z|}^{p}-c\le f\left(x,z\right)\le L\left(1+{|z|}^{q}\right).$(1.2)

In the scalar case $N=1$, when $p=q$, the local minimizers $u\in {W}^{1,p}\left(\mathrm{\Omega }\right)$ of (1.1) are locally Hölder continuous, see [13] and [19, p. 361]. If $p and q is far from p, then the local minimizers might be unbounded, see [12, 17, 18, 15] and [19, Section 5]. We are concerned with higher integrability for the gradient of minimizers. More precisely, assume that $u:\mathrm{\Omega }\subset {ℝ}^{n}\to {ℝ}^{N}$ makes the energy (1.1) finite. Then the left-hand side of (1.2) implies that $Du\in {L}^{p}$. In addition to finite energy, we assume that u is a minimizer of (1.1) and we ask: Does the minimality of u boost the integrability of the gradient Du from ${L}^{p}$ to ${L}^{q}$? The answer is given in [9]: We assume that (1.2) holds with $\nu =1$, $c=0$, we require that $z\to f\left(x,z\right)\in {C}^{1}\left({ℝ}^{nN}\right)$ and, for constants $\mu \in \left[0,1\right]$ and $\alpha \in \left(0,1\right]$, we assume that the following hold:

${L}^{-1}{\left({\mu }^{2}+{|{z}_{1}|}^{2}+{|{z}_{2}|}^{2}\right)}^{\frac{p-2}{2}}{|{z}_{1}-{z}_{2}|}^{2}\le 〈\frac{\partial f}{\partial z}\left(x,{z}_{1}\right)-\frac{\partial f}{\partial z}\left(x,{z}_{2}\right);{z}_{1}-{z}_{2}〉,$(1.3)$|\frac{\partial f}{\partial z}\left(x,z\right)-\frac{\partial f}{\partial z}\left(y,z\right)|\le L|x-y|{}^{\alpha }\left(1+|z{|}^{q-1}\right).$(1.4)

The exponents p and q must be close enough, i.e.,

$1(1.5)

We recall that n is the dimension of the space where x lives, i.e., $x\in \mathrm{\Omega }\subset {ℝ}^{n}$. Let us remark that (1.5) asserts that the smaller α is, the closer p and q must be. In addition, we assume that the Lavrentiev gap on u is zero:

$\mathcal{ℒ}\left(u,{B}_{R}\right)=0$(1.6)

for every ball ${B}_{R}\subset \subset \mathrm{\Omega }$. Such a Lavrentiev gap will be defined in the next section. Under (1.2)–(1.6), the local minimizers u of (1.1) enjoy higher integrability, i.e., $Du\in {L}_{\mathrm{loc}}^{q}\left(\mathrm{\Omega }\right).$ Checking (1.6) is not easy for non-autonomous densities $f\left(x,z\right)$; it has been done in [9] for some model functionals using some arguments due to [22]. The aim of the present paper is to give a sufficient condition on the density $f\left(x,z\right)$ that ensures the vanishing of the Lavrentiev gap (1.6), see Theorem 3.1(4).

## 2 Preliminaries

In the following Ω will be an open, bounded subset of ${ℝ}^{n}$, $n\ge 2$, and we will denote

${B}_{R}\equiv {B}_{R}\left({x}_{0}\right):=\left\{x\in {ℝ}^{n}:|x-{x}_{0}|

where, unless differently specified, all the balls considered will have the same center. We assume that $f:\mathrm{\Omega }×{ℝ}^{nN}\to ℝ$ is a Caratheodory function verifying the $\left(p,q\right)$-growth (1.2) with $\nu =1$ and $c=0$. Moreover, we assume that $z\to f\left(x,z\right)$ is convex. Due to the non-standard growth behavior of f, we shall adopt the following notion of local minimizer.

#### Definition 2.1.

A function $u\in {W}_{\mathrm{loc}}^{1,1}\left(\mathrm{\Omega };{ℝ}^{N}\right)$ is a local minimizer of $\mathcal{ℱ}$ if and only if $x↦f\left(x,Du\left(x\right)\right)\in {L}_{\mathrm{loc}}^{1}\left(\mathrm{\Omega }\right)$ and

${\int }_{\mathrm{supp}\varphi }f\left(x,Du\left(x\right)\right)𝑑x\le {\int }_{\mathrm{supp}\varphi }f\left(x,Du\left(x\right)+D\varphi \left(x\right)\right)𝑑x,$

for any $\varphi \in {W}^{1,1}\left(\mathrm{\Omega };{ℝ}^{N}\right)$ with $\mathrm{supp}\varphi \subset \subset \mathrm{\Omega }$.

Let us now explain the Lavrentiev gap. We adopt the viewpoint of [4], see also [3]. Let us set

$X={W}^{1,p}\left({B}_{R};{ℝ}^{N}\right),Y={W}_{\mathrm{loc}}^{1,q}\left({B}_{R};{ℝ}^{N}\right)\cap {W}^{1,p}\left({B}_{R};{ℝ}^{N}\right).$

We consider functionals $\mathcal{𝒢}:X\to \left[0,+\mathrm{\infty }\right]$ that are sequentially weakly lower semicontinuous (s.w.l.s.c.) on X, and we set

We have ${\overline{\mathcal{ℱ}}}_{X}\le {\overline{\mathcal{ℱ}}}_{Y}$, and we define the Lavrentiev gap as follows:

when ${\overline{\mathcal{ℱ}}}_{X}\left(v\right)<+\mathrm{\infty }$ and $\mathcal{ℒ}\left(v,{B}_{R}\right)=0$ if ${\overline{\mathcal{ℱ}}}_{X}\left(v\right)=+\mathrm{\infty }$. Since $f\left(x,z\right)$ is convex with respect to z, standard weak lower semicontinuity results give ${\overline{\mathcal{ℱ}}}_{X}=\mathcal{ℱ}$ (see, for instance, [14, Chapter 4]).

The following lemma will be used in the proof of the main theorem (see [4]).

#### Lemma 2.2.

Let $u\mathrm{\in }{W}^{\mathrm{1}\mathrm{,}p}\mathit{}\mathrm{\left(}{B}_{R}\mathrm{;}{\mathrm{R}}^{N}\mathrm{\right)}$ be a function such that $\mathcal{F}\mathit{}\mathrm{\left(}u\mathrm{,}{B}_{R}\mathrm{\right)}\mathrm{<}\mathrm{+}\mathrm{\infty }$. Then $\mathcal{L}\mathit{}\mathrm{\left(}u\mathrm{,}{B}_{R}\mathrm{\right)}\mathrm{=}\mathrm{0}$ if and only if there exists a sequence ${\mathrm{\left\{}{u}_{m}\mathrm{\right\}}}_{m\mathrm{\in }\mathrm{N}}\mathrm{\subset }{W}_{\mathrm{loc}}^{\mathrm{1}\mathrm{,}q}\mathit{}\mathrm{\left(}{B}_{R}\mathrm{;}{\mathrm{R}}^{N}\mathrm{\right)}\mathrm{\cap }{W}^{\mathrm{1}\mathrm{,}p}\mathit{}\mathrm{\left(}{B}_{R}\mathrm{;}{\mathrm{R}}^{N}\mathrm{\right)}$ such that

and

$\mathcal{ℱ}\left({u}_{m},{B}_{R}\right)\to \mathcal{ℱ}\left(u,{B}_{R}\right).$

## 3 Main section

#### Theorem 3.1.

Let $f\mathrm{:}\mathrm{\Omega }\mathrm{×}{\mathrm{R}}^{n\mathit{}N}\mathrm{\to }\mathrm{R}$ be a function satisfying the following conditions:

• (1)

${|z|}^{p}\le f\left(x,z\right)\le L\left(1+{|z|}^{q}\right)$, $1,

• (2)

$|f\left(x,z\right)-f\left(\stackrel{~}{x},z\right)|\le H{|x-\stackrel{~}{x}|}^{\alpha }\left(1+{|z|}^{q}\right)$, $0<\alpha \le 1$,

• (3)

$z↦f\left(x,z\right)$ is convex for all x,

• (4)

for ${B}_{R}\subset \subset \mathrm{\Omega }$, ${\epsilon }_{0}\in \left(0,1\right]$ such that ${B}_{R+2{\epsilon }_{0}}\subset \subset \mathrm{\Omega }$, $x\in {B}_{R}$ and $\epsilon \in \left(0,{\epsilon }_{0}\right)$ , there exists $\stackrel{~}{y}=\stackrel{~}{y}\left(x,\epsilon \right)\in \overline{B\left(x,\epsilon \right)}$ such that for $z\in {ℝ}^{nN}$ and $y\in \overline{B\left(x,\epsilon \right)}$ , we have $f\left(\stackrel{~}{y},z\right)\le f\left(y,z\right)$.

Let $u\mathrm{\in }{W}_{\mathrm{loc}}^{\mathrm{1}\mathrm{,}p}\mathit{}\mathrm{\left(}\mathrm{\Omega }\mathrm{;}{\mathrm{R}}^{N}\mathrm{\right)}$ such that $x\mathrm{↦}f\mathit{}\mathrm{\left(}x\mathrm{,}D\mathit{}u\mathit{}\mathrm{\left(}x\mathrm{\right)}\mathrm{\right)}\mathrm{\in }{L}_{\mathrm{loc}}^{\mathrm{1}}\mathit{}\mathrm{\left(}\mathrm{\Omega }\mathrm{\right)}$ and assume that

$q\le p\left(\frac{n+\alpha }{n}\right).$(3.1)

Then $\mathcal{L}\mathit{}\mathrm{\left(}u\mathrm{,}{B}_{R}\mathrm{\right)}\mathrm{=}\mathrm{0}$ for all ${B}_{R}\mathrm{\subset }\mathrm{\subset }\mathrm{\Omega }$.

#### Remark 3.2.

Let us now explain condition (4). For every fixed z, $y\to f\left(y,z\right)$ is continuous, so the minimization of $f\left(y,z\right)$ when $y\in \overline{B\left(x,\epsilon \right)}$ gives a minimizer y depending on $x,\epsilon$ and z. Condition (4) asks for independence on z, i.e., there exists a minimizer $\stackrel{~}{y}$ that works for every z. We will first give the proof of Theorem 3.1, and then we will show examples of densities $f\left(x,z\right)$ satisfying condition (4).

#### Remark 3.3.

Let us compare (1.5) with (3.1). When proving the absence of the Lavrentiev gap $\mathcal{ℒ}\left(u,{B}_{R}\right)=0$, the borderline case $q=p\left(\frac{n+\alpha }{n}\right)$ is allowed but we need strict inequality (1.5) when proving higher integrability of minimizers, see [9, p. 32].

#### Proof.

Consider $0<\epsilon <{\epsilon }_{0}\le 1$ as in hypothesis (4), then $u\in {W}^{1,p}\left({B}_{R+2{\epsilon }_{0}};{ℝ}^{N}\right)$ and

$\mathcal{ℱ}\left(u,{B}_{R+2{\epsilon }_{0}}\right)={\int }_{{B}_{R+2{\epsilon }_{0}}}f\left(x,Du\left(x\right)\right)𝑑x<+\mathrm{\infty }.$

Let us denote ${u}_{\epsilon }\left(x\right):=\left(u*{\varphi }_{\epsilon }\right)\left(x\right)$, the usual mollification, where $x\in {B}_{R}$, and define

${f}_{\epsilon }\left(x,z\right)=\underset{y\in \overline{B\left(x,\epsilon \right)}}{\mathrm{min}}f\left(y,z\right).$(3.2)

By definition, it follows that

$|D{u}_{\epsilon }\left(x\right)|\le \left({\int }_{{B}_{R+2{\epsilon }_{0}}}{|Du\left(y\right){|}^{p}dy\right)}^{\frac{1}{p}}\left({\int }_{{ℝ}^{n}}{|{\varphi }_{\epsilon }\left(y\right){|}^{{p}^{\prime }}\right)}^{\frac{1}{{p}^{\prime }}}\le C{\epsilon }^{-\frac{n}{p}},$

where $C=C\left({\parallel Du\parallel }_{{L}^{p}}\right)>1$. Moreover, by the Hölder continuity hypothesis (i.e., hypothesis 2), we have

${f}_{\epsilon }\left(x,z\right)\ge f\left(x,z\right)-H{\epsilon }^{\alpha }\left(1+{|z|}^{q}\right).$(3.3)

Note that the left-hand side of hypothesis (1) gives

${|z|}^{p}\le {f}_{\epsilon }\left(x,z\right).$(3.4)

Now we observe that is possible to find $K=K\left(p,q,{\parallel Du\parallel }_{{L}^{p}},H\right)<+\mathrm{\infty }$ such that

$f\left(x,z\right)\le K{f}_{\epsilon }\left(x,z\right)+H,x\in {B}_{R},|z|\le C{\epsilon }^{-\frac{n}{p}}.$(3.5)

Indeed, let us fix $\delta \in \left(0,1\right)$ and observe that, using (3.3), (3.4) and $|z|\le C{\epsilon }^{-\frac{n}{p}}$, we get

${f}_{\epsilon }\left(x,z\right)=\delta {f}_{\epsilon }\left(x,z\right)+\left(1-\delta \right){f}_{\epsilon }\left(x,z\right)$$\ge \delta f\left(x,z\right)-\delta H{\epsilon }^{\alpha }\left(1+{|z|}^{q}\right)+\left(1-\delta \right){|z|}^{p}$$=\delta f\left(x,z\right)-\delta H{\epsilon }^{\alpha }{|z|}^{q}+\left(1-\delta \right){|z|}^{p}-\delta H{\epsilon }^{\alpha }$$=\delta f\left(x,z\right)-\delta H{\epsilon }^{\alpha }{|z|}^{p}{|z|}^{q-p}+\left(1-\delta \right){|z|}^{p}-\delta H{\epsilon }^{\alpha }$$\ge \delta f\left(x,z\right)-\delta {C}^{q-p}H{\epsilon }^{\alpha +\left(\frac{p-q}{p}\right)n}{|z|}^{p}+\left(1-\delta \right){|z|}^{p}-\delta H{\epsilon }^{\alpha }$$\ge \delta f\left(x,z\right)+\left(1-\delta -\delta {C}^{q-p}H\right){|z|}^{p}-\delta H,$

where the last estimate relies on the fact that $\frac{q}{p}\le \frac{n+\alpha }{n}$, $0<\alpha \le 1$ and $0<\epsilon <1$. Then (3.5) follows choosing $K=\frac{1}{\delta }=1+{C}^{q-p}H$. Now, using hypothesis (4), Jensen’s inequality and (3.2), we obtain

${f}_{\epsilon }\left(x,D{u}_{\epsilon }\left(x\right)\right)=f\left(\stackrel{~}{y},D{u}_{\epsilon }\left(x\right)\right)$$\le {\int }_{B\left(x,\epsilon \right)}f\left(\stackrel{~}{y},Du\left(y\right)\right){\varphi }_{\epsilon }\left(x-y\right)𝑑y$$\le {\int }_{B\left(x,\epsilon \right)}f\left(y,Du\left(y\right)\right){\varphi }_{\epsilon }\left(x-y\right)𝑑y$$=\left(f\left(\cdot ,Du\left(\cdot \right)\right)*{\varphi }_{\epsilon }\right)\left(x\right)$$=:f{\left(\cdot ,Du\left(\cdot \right)\right)}_{\epsilon }\left(x\right).$(3.6)

Therefore, using (3.5), we have

$f\left(x,D{u}_{\epsilon }\left(x\right)\right)\le Kf{\left(\cdot ,Du\left(\cdot \right)\right)}_{\epsilon }\left(x\right)+H.$

Finally, since $f{\left(\cdot ,Du\left(\cdot \right)\right)}_{\epsilon }\left(x\right)\to f\left(x,Du\left(x\right)\right)$ strongly in ${L}^{1}\left({B}_{R}\right)$, by recalling that ${u}_{\epsilon }\to u$ in ${W}^{1,p}\left({B}_{R};{ℝ}^{N}\right)$, and by using a well-known variant of Lebesgue’s dominated convergence theorem and Lemma 2.2, the proof is completed. ∎

#### Remark 3.4.

We note that our assumption (4) is very close to [22, assumption (2.3)]. Our proof is inspired by the one of [9, Lemma 13], which, in turn, is based on some arguments used in [22].

#### Remark 3.5.

Now we give some examples of functions for which Theorem 3.1 is valid.

• (1)

$f\left(x,z\right)=b\left(z\right)+a\left(x\right)c\left(z\right)$ with the following conditions:

• (1)(i)

$a\in {C}^{0,\alpha }\left(\overline{\mathrm{\Omega }}\right)$ and $a\left(x\right)\ge 0$ for all x,

• (1)(ii)

b and c are convex functions such that

For instance, we can consider the following functions:

• $f\left(x,z\right)=b\left(z\right)$, independent of x.

• $f\left(x,z\right)={|z|}^{p}+a\left(x\right){|z|}^{q}$. This example has been already dealt with in [9]; see also [22, 10, 8, 7, 1, 6].

• $f\left(x,z\right)={|z|}^{p}+a\left(x\right){|z|}^{p}\mathrm{ln}\left(e+|z|\right)$. This example is taken from [2, 1].

• $f\left(x,z\right)={|z|}^{2}+a\left(x\right){\left[\mathrm{max}\left\{{z}_{n},0\right\}\right]}^{q}$, where $q>2$. This example is inspired by [21].

• $f\left(x,z\right)={|z|}^{p}+a\left(x\right)\left[{|{z}_{1}-{z}_{2}|}^{q}+{|{z}_{1}|}^{q}\right]$, where $1. This example is inspired by [5].

• (2)

$f\left(x,z\right)={\sum }_{i=1}^{k}\left[{b}_{i}\left(z\right)+{a}_{i}\left(x\right){c}_{i}\left(z\right)\right]$, where $k\in ℕ$ and ${a}_{i},{b}_{i},{c}_{i}$ verify the corresponding conditions of the previous example for all $i\in \left\{1,2,\mathrm{\dots },k\right\}$.

• (3)

$f\left(x,z\right)=h\left({\sum }_{i=1}^{k}\left[{b}_{i}\left(z\right)+{a}_{i}\left(x\right){c}_{i}\left(z\right)\right]\right)$, where, in addition to the previous conditions, h is increasing, convex, Lipschitz and such that $s\le h\left(s\right)\le \alpha s+\beta$.

• (4)

$f\left(x,z\right)=h\left(a\left(x\right),z\right)$ with the following conditions:

• (4)(i)

$t↦h\left(t,z\right)$ is increasing,

• (4)(ii)

h is convex with respect to the second variable,

• (4)(iii)

$a\in C\left(\overline{\mathrm{\Omega }}\right)$,

• (4)(iv)

f verifies assumptions (1) and (2) of Theorem 3.1.

For example,

$f\left(x,z\right)={|z|}^{p}+{\left(e+\stackrel{~}{a}\left(x\right)|z|\right)}^{a+b\mathrm{sin}\left(\mathrm{ln}\left(\mathrm{ln}\left(e+\stackrel{~}{a}\left(x\right)|z|\right)\right)\right)},$

where $\stackrel{~}{a}\in {C}^{0,\alpha }\left(\overline{\mathrm{\Omega }}\right)$, $\stackrel{~}{a}\left(x\right)\ge 0$ for all x, $a\ge 1+2b\sqrt{2}$ and $b>0$. In order to satisfy the non-standard $\left(p,q\right)$-growth condition, we can consider $1. This example is inspired by [11, 20], see also [16].

#### Remark 3.6.

Hypothesis (4) was used during the proof of Theorem 3.1 in order to obtain the second increase in (3.6). Now we want to show an example of a function for which hypothesis (4) fails. Let us consider $\mathrm{\Omega }=B\left(0,1\right)\subset {ℝ}^{2}$ and the function $f:B\left(0,1\right)×{ℝ}^{2N}\to ℝ$ such that

$f\left(x,z\right)={|z|}^{p}+a\left(x\right)\left({|z|}^{q}-1\right)+1,$

where, for $x=\left({x}_{1},{x}_{2}\right)$,

In this case the minimum point of the function changes depending on the choice of z. Indeed, let us consider ${B}_{R}=B\left(0,\frac{1}{2}\right)$, ${\epsilon }_{0}=\frac{1}{8}$, $x=0$. Then we deal with the two cases: $|z|=0$ and $|z|=2$.

When $|z|=0$, we have

and then the minimum value in $\overline{B\left(0,\epsilon \right)}$ is reached for $\stackrel{~}{y}=\left(0,\epsilon \right)$.

If $|z|=2$, then

and therefore, in this situation, $\stackrel{~}{y}$ is any point $\left({y}_{1},{y}_{2}\right)$ such that ${y}_{2}\le 0$.

#### Corollary 3.7.

Let $h\mathrm{:}\mathrm{\Omega }\mathrm{×}{\mathrm{R}}^{n\mathit{}N}\mathrm{\to }\mathrm{R}$ be a function verifying Theorem 3.1 and let $f\mathrm{:}\mathrm{\Omega }\mathrm{×}{\mathrm{R}}^{n\mathit{}N}\mathrm{\to }\mathrm{R}$ be a function such that

$h\left(x,z\right)-{c}_{1}\le f\left(x,z\right)\le h\left(x,z\right)+{c}_{2},$

where ${c}_{\mathrm{1}}\mathrm{,}{c}_{\mathrm{2}}\mathrm{\ge }\mathrm{0}$. We consider $u\mathrm{\in }{W}_{\mathrm{loc}}^{\mathrm{1}\mathrm{,}p}\mathit{}\mathrm{\left(}\mathrm{\Omega }\mathrm{;}{\mathrm{R}}^{N}\mathrm{\right)}$ such that $x\mathrm{↦}f\mathit{}\mathrm{\left(}x\mathrm{,}D\mathit{}u\mathit{}\mathrm{\left(}x\mathrm{\right)}\mathrm{\right)}\mathrm{\in }{L}_{\mathrm{loc}}^{\mathrm{1}}\mathit{}\mathrm{\left(}\mathrm{\Omega }\mathrm{\right)}$ and assume $q\mathrm{\le }p\mathit{}\mathrm{\left(}\frac{n\mathrm{+}\alpha }{n}\mathrm{\right)}$. Then $\mathcal{L}\mathit{}\mathrm{\left(}u\mathrm{,}{B}_{R}\mathrm{\right)}\mathrm{=}\mathrm{0}$ for all ${B}_{R}\mathrm{\subset }\mathrm{\subset }\mathrm{\Omega }$.

#### Proof.

We follow the proof of [9, Theorem 6]. By the proof of Theorem 3.1, if $u\in {W}_{\mathrm{loc}}^{1,p}\left(\mathrm{\Omega };{ℝ}^{N}\right)$ is such that $h\left(x,Du\left(x\right)\right)\in {L}_{\mathrm{loc}}^{1}\left(\mathrm{\Omega }\right)$, then there exists a sequence ${\left\{{u}_{m}\right\}}_{m\in ℕ}\subset {W}^{1,q}\left({B}_{R};{ℝ}^{N}\right)$ such that ${u}_{m}\to u$ strongly in ${W}^{1,p}\left({B}_{R};{ℝ}^{N}\right)$, $D{u}_{m}\left(x\right)\to Du\left(x\right)$ a.e., $h\left(x,D{u}_{m}\left(x\right)\right)\to h\left(x,Du\left(x\right)\right)$ a.e., and ${\int }_{{B}_{R}}h\left(x,D{u}_{m}\left(x\right)\right)\to {\int }_{{B}_{R}}h\left(x,Du\left(x\right)\right)$. Using a well-known variant of Lebesgue’s dominated convergence theorem, we have that

${\int }_{{B}_{R}}f\left(x,D{u}_{m}\left(x\right)\right)\to {\int }_{{B}_{R}}f\left(x,Du\left(x\right)\right),$

and then, by Lemma 2.2, $\mathcal{ℒ}\left(u,{B}_{R}\right)=0$. ∎

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## About the article

Received: 2016-09-10

Accepted: 2016-09-27

Published Online: 2016-12-20

The authors thank GNAMPA and UNIVAQ for the support.

Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 73–78, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496,

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© 2019 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0

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