Existence of renormalized solutions to elliptic equation in Musielak-Orlicz space

TL;DR

This paper establishes the existence of renormalized solutions for a broad class of nonlinear elliptic equations in Musielak-Orlicz spaces without growth restrictions, using advanced approximation and measure techniques.

Contribution

It proves the existence of solutions in Musielak-Orlicz spaces under minimal growth assumptions, extending previous results to more general anisotropic and nonhomogeneous settings.

Findings

Existence of renormalized solutions without growth restrictions.

Applicability to general nonlinear elliptic equations in Musielak-Orlicz spaces.

Use of truncation, Young measures, and monotonicity methods.

Abstract

We prove existence of renormalized solutions to general nonlinear elliptic equation in Musielak-Orlicz space avoiding growth restrictions. Namely, we consider \begin{equation*} -{\rm div} A(x,\nabla u)= f\in L^1(\Omega), \end{equation*} on a Lipschitz bounded domain in $\mathbb{R}^N$. The growth of the monotone vector field $A$ is controlled by a generalized nonhomogeneous and anisotropic $N$-function $M $. The approach does not require any particular type of growth condition of $M$ or its conjugate $M^*$ (neither $\Delta_2$, nor $\nabla_2$). The condition we impose is log-Holder continuity of $M$, which results in good approximation properties of the space. The proof of the main results uses truncation ideas, the Young measures methods and monotonicity arguments.

Equations

$$-\mathrm{div}A(x,\nabla u)=f\in L^1(\Omega ),$$

$$\left\{\begin{array}[]{cl}-\mathrm{div}A(x,\nabla u)=f&\qquad\mathrm{in}\qquad\Omega,\\ u(x)=0&\qquad\mathrm{on}\qquad\partial\Omega,\end{array}\right.$$

$$A(x,\xi )\xi \ge c_A\left(M(x,\xi )+M^{\ast }(x,A(x,\xi ))\right),$$

$$(A(x,\xi )-A(x,\eta ))\cdot (\xi -\eta )\ge 0.$$

$$V_0^M=\{u\in W_0^{1,1}(\Omega ):\nabla u\in L_M(\Omega ;{\mathbb{R}}^N)\}.$$

$${\int }_{G}M(x,z)dx<\infty .$$

$$\frac{M(x,\xi )}{(M_j^{\delta }(\xi ){)}^{\ast \ast }}\le c\left(1+|\xi {|}^{-\frac{a}{\log (b\delta )}}\right),$$

$$M_j^{\delta }(\xi ):=\underset{x\in {\tilde{Q}}_j^{\delta }\cap \Omega }{\inf }M(x,\xi ),$$

$$T_{k}(f)(x)=\left\{\begin{array}[]{ll}f&|f|\leq k,\\ k\frac{f}{|f|}&|f|\geq k.\end{array}\right.$$

$$T_k(u)\in V_0^M\cap L^{\infty }(\Omega ),A(x,\nabla T_k(u))\in L_{M^{\ast }}(\Omega ;{\mathbb{R}}^N).$$

$${\int }_{\Omega }A(x,\nabla u)\cdot \nabla (h(u)\phi )dx={\int }_{\Omega }fh(u)\phi dx.$$

$$\left\{\begin{array}[]{cl}-\mathrm{div}A(x,\nabla u)=f&\qquad\mathrm{in}\qquad\Omega,\\ u(x)=0&\qquad\mathrm{on}\qquad\partial\Omega,\end{array}\right.$$

$$A(x,\xi )=\alpha (x)|\xi {|}^{p(x)-2}\xi \text{ with }0<<\alpha (x)\in L^{\infty }(\Omega )\cap C(\Omega );$$

$$A(x,\xi )=\sum _{i=1}^N{\alpha }_i(x)|\xi {|}^{p_i(x)-2}\xi \text{ with }0<<{\alpha }_i(x)\in L^{\infty }(\Omega )\cap C(\Omega ).$$

$$M(x,\xi )=\sum _{i=1}^j{k}_i(x)M_i(\xi )+M_0(x,|\xi |),j\in \mathbb{N},$$

$$c_A\left(M(x,\xi )+M^{\ast }(x,A(x,\xi ))\right)\le A(x,\xi )\xi \le M(x,\xi )+M^{\ast }(x,A(x,\xi ))$$

$$A(x,\xi )\xi \ge c_A\left(M(x,\xi )+M^{\ast }(x,A(x,\xi ))\right)-k(x),0\le k(x)\in L^1(\Omega ),$$

$$\frac{M(x,\xi )}{M(y,\xi )}\le \max \left\{|\xi {|}^{-\frac{a_1}{\log |x-y|}},b_1^{-\frac{a_1}{\log |x-y|}}\right\},\text{with some}{a}_1>0,b_1\ge 1.$$

$$|p(x)-p(y)|\le \frac{a}{\log \left(\frac{1}{|x-y|}\right)}.$$

$$\frac{M(x,\xi )}{M(y,\xi )}=\frac{|\xi {|}^{p(x)}}{|\xi {|}^{p(y)}}=|\xi {|}^{p(x)-p(y)}\le |\xi {|}^{\frac{a}{\log \left(\frac{1}{|x-y|}\right)}}=|\xi {|}^{-\frac{a}{\log |x-y|}}.$$

$${\int }_{\Omega }M(x,\xi (x))dx<\infty .$$

$$||\xi |{|}_{L_M}=\inf \left\{\lambda >0:{\int }_{\Omega }M\left(x,\frac{\xi (x)}{\lambda }\right)dx\le 1\right\}.$$

$$E_M(\Omega ;{\mathbb{R}}^N)\subset {\mathcal{L}}_M(\Omega ;{\mathbb{R}}^N)\subset L_M(\Omega ;{\mathbb{R}}^N),$$

$$E_M(\Omega ;{\mathbb{R}}^N)={\mathcal{L}}_M(\Omega ;{\mathbb{R}}^N)=L_M(\Omega ;{\mathbb{R}}^N)$$

$$L^{\infty }(\Omega ;{\mathbb{R}}^N)\subset L_M(\Omega ;{\mathbb{R}}^N).$$

$$A(x,\xi )\xi \ge c_A|\xi {|}^p$$

$${\tilde{c}}_A|\xi {|}^{p-1}\ge |A(x,\xi )|.$$

$$A(x,\xi )\xi \ge c_{A}M(x,\xi ),$$

$$P(s):=\underset{\xi :|\xi |=s}{\sup }{\left(\underset{x\in \Omega ,}{\inf }{M}^{\ast }(x,\xi )\right)}^{\ast }.$$

$$|A(x,\xi )|\le 2(P^{\ast }{)}^{-1}\left(\frac{1}{c_A}P\left(\frac{2}{c_A}|\xi |\right)\right).$$

Full text

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on a Lipschitz bounded domain in ${\mathbb{R}^{N}}$. The growth of the monotone vector field $A$ is controlled by a generalized nonhomogeneous and anisotropic $N$-function $M$. The approach does not require any particular type of growth condition of $M$ or its conjugate $M^{*}$ (neither $\Delta_{2}$, nor $\nabla_{2}$). The condition we impose is log-Hölder continuity of $M$, which results in good approximation properties of the space. The proof of the main results uses truncation ideas, the Young measures methods and monotonicity arguments.

Key words and phrases: elliptic problems, existence of solutions, Musielak-Orlicz spaces, renormalized solutions

Mathematics Subject Classification (2010): 35J60, 35D30.

1 Introduction

Our aim is to find a way of proving the existence of renormalized solutions to a strongly nonlinear elliptic equation with $L^{1}$-data under minimal restrictions on the growth of the leading part of the operator. We investigate operators $A$, which are monotone, but not necessarily strictly. The modular function $M$, which controls the growth of the operator, is not assumed to be isotropic, i.e. $M=M(x,\xi)$ not $M=M(x,|\xi|)$. In turn, we can expect different behaviour of $M(x,\cdot)$ in various directions. We do not require $M\in\Delta_{2}$, nor $M^{*}\in\Delta_{2}$, nor any particular growth of $M$, such as $M(x,\xi)\geq c|\xi|^{1+\nu}$ for $\xi>\xi_{0}$. The price we pay for relaxing the conditions on the growth is requirement of log-Hölder-type regularity of the modular function (cf. condition (M)).

We study the problem

[TABLE]

where $\Omega$ is a bounded Lipschitz domain in ${\mathbb{R}^{N}}$, $N>1$, $f:\Omega\to{\mathbb{R}}$, $f\in L^{1}(\Omega)$.

We consider $A$ belonging to an Orlicz class with respect to the second variable. Namely, we assume that function $A:\Omega\times{\mathbb{R}^{N}}\to{\mathbb{R}^{N}}$ satisfies the following conditions.

• (A1)

$A$ is a Carathéodory’s function;

• (A2)

There exists an $N$-function $M:\Omega\times{\mathbb{R}^{N}}\to{\mathbb{R}}$ and a constant $c_{A}\in(0,1]$ such that for all $\xi\in{\mathbb{R}^{N}}$ we have

[TABLE]

where $M^{*}$ is conjugate to $M$ (see Definition 3.2);

• (A3)

For all $\xi,\eta\in{\mathbb{R}^{N}}$ and $x\in\Omega$ we have

[TABLE]

Existence of solutions to (1) is considered in

[TABLE]

The space $L_{M}$ (Definition 2.1) is equipped with the modular function $M$ being an $N$-function (Definition 3.1) controlling the growth of $A$, cf. (A2).

Unlike other studies, instead of growth conditions we assume regularity of $M$.

• (M)

Suppose for every measurable set $G\subset\Omega$ and every $z\in{\mathbb{R}^{N}}$ we have

[TABLE]

Let us consider a family of $N$-dimensional cubes covering the set $\Omega$. Namely, a family $\{{Q_{j}^{\delta}}\}_{j=1}^{N_{\delta}}$ consists of closed cubes of edge $2\delta$, such that $\mathrm{int}{Q_{j}^{\delta}}\cap\mathrm{int}Q^{\delta}_{i}=\emptyset$ for $i\neq j$ and $\Omega\subset\bigcup_{j=1}^{N_{\delta}}{Q_{j}^{\delta}}$. Moreover, for each cube ${Q_{j}^{\delta}}$ we define the cube ${\widetilde{Q}_{j}^{\delta}}$ centered at the same point and with parallel corresponding edges of length $4\delta$. Assume that there exist constants $a,b,c,\delta_{0}>0$, such that for all $\delta<\delta_{0}$, $x\in{Q_{j}^{\delta}}$ and all $\xi\in{\mathbb{R}^{N}}$ we have

[TABLE]

where

[TABLE]

while ${({M_{j}^{\delta}}(\xi))^{**}}=({({M_{j}^{\delta}}(\xi))}^{*})^{*}$ is the greatest convex minorant of ${M_{j}^{\delta}}(\xi)$ (coinciding with the second conjugate cf. Definition 3.2).

In further parts of the paper we describe the cases, when the above condition is not necessary. Let us only point out that to get (M) in the isotropic case, i.e. when we consider $M(x,\xi)=M(x,|\xi|)$, it suffices to assume log-Hölder-type condition with respect to $x$ (6), cf. Lemma 3.2.

We apply the truncation techniques. Let truncation $T_{k}(f)(x)$ be defined as follows

[TABLE]

We call a function $u$ a renormalized solution to (1), when it satisfies the following conditions.

• (R1)

$u:\Omega\to{\mathbb{R}}$ is measurable and for each $k>0$

[TABLE]

• (R2)

For every $h\in C^{1}_{c}({\mathbb{R}})$ and all $\varphi\in V_{0}^{M}\cap L^{\infty}(\Omega)$ we have

[TABLE]

• (R3)

$\int_{\{l<|u|<l+1\}}A(x,\nabla u)\cdot\nabla u\,dx\to 0$ as $l\to\infty$.

Our main result reads as follows.

Theorem 1.1**.**

Suppose $f\in L^{1}(\Omega)$, an $N$-function $M$ satisfies assumption (M) and function $A$ satisfies assumptions (A1)-(A3). Then there exists at least one renormalized weak solution to the problem

[TABLE]

Namely, there exists $u\in V_{0}^{M}$ which satisfies (R1)-(R3). Moreover, $A(\cdot,\nabla u)\in L_{M^{∗}}(\Omega)$.

Example 1.1**.**

We give below pairs of functions $M$ and $A$ satisfying conditions (M) and (A1)-(A3), respectively.

• •

Consider $M(x,\xi)=|\xi|^{p(x)}$ with log-Hölder $p:\Omega\to[p^{-},p^{+}]$, where $p^{-}=\inf_{x\in\Omega}p(x)>1$ and $p^{+}=\sup_{x\in\Omega}p(x)<\infty,$ then $V_{0}^{M}=W_{0}^{1,p(\cdot)}(\Omega)$ and we admitt $A(x,\xi)=|\xi|^{p(x)-2}\xi$ ($p(\cdot)$-Laplacian case) as well as

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• •

$M(x,\xi)=\sum_{i=1}^{N}|\xi_{i}|^{p_{i}(x)}$, where $\xi=(\xi_{1},\dots,\xi_{N})\in{\mathbb{R}^{N}}$, log-Hölder functions $p_{i}:\Omega\to[p_{i}^{-},p_{i}^{+}]$, $i=1,\dots,N$, where $p_{i}^{-}=\inf_{x\in\Omega}p_{i}(x)>1$ and $p_{i}^{+}=\sup_{x\in\Omega}p_{i}(x)<\infty,$ then $V_{0}^{M}=W_{0}^{1,\vec{p}(\cdot)}(\Omega)$ and we admitt

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Remark 1.1** (cf. [9]).**

When the modular function has a special form we can simplify our assumptions. In the case of $M(x,\xi)=M(x,|\xi|)$, via Lemma 3.2, we replace condition (M) in the above theorem by log-Hölder continuity of M, cf. (6). If $M$ has a form

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instead of whole (M) we assume only that $M_{0}$ is log-Hölder continuous (6), all $M_{i}$ for $i=1,\dots,j$ are $N$-functions and all $k_{i}$ are nonnegative and satisfy $\frac{k_{i}(x)}{k_{i}(y)}\leq C_{i}^{\log\frac{1}{|x-y|}}$ with $C_{i}>0$ for $i=1,\dots,j$.

Remark 1.2**.**

Note that according to (A2) and the Fenchel-Young inequality we have

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satisfied with a certain $N$-function $M:\Omega\times{\mathbb{R}^{N}}\to{\mathbb{R}}$. This observation results in $A(x,0)=0$. However, the framework admitts considering in (A2)

[TABLE]

despite it does not imply $A(x,0)=0$.

The Musielak-Orlicz spaces equipped with the modular function satisfying $\Delta_{2}$-condition (cf. Definition 3.3) have strong properties, however there is a vast range of $N$-functions not satisfying it, e.g.

• i)

$M(x,\xi)=a(x)\left(\exp(|\xi|)-1+|\xi|\right)$;

• ii)

$M(x,\xi)=|\xi_{1}|^{p_{1}(x)}\left(1+|\log|\xi||\right)+\exp(|\xi_{2}|^{p_{2}(x)})-1$, when $(\xi_{1},\xi_{2})\in{\mathbb{R}}^{2}$ and $p_{i}:\Omega\to[1,\infty]$. It is a model example to imagine what we mean by anisotropic modular function.

Nonetheless, our assumption that $M,M^{*}$ are $N$-functions (Definition 3.1) in the variable exponent setting restrict us to the case of $1<p_{-}\leq p(x)\leq p^{+}<\infty$.

State of art

Existence to problems like (1) is very well understood, when $A$ is independent of the spacial variable and has a polynomial growth. In particular, there is vast literature for analysis of the case involving the $p$-Laplace operator $A(x,\xi)=|\xi|^{p-2}\xi$ and problems stated in the Lebesgue space setting (the modular function is then $M(x,\xi)=|\xi|^{p}$). Let us note that the variable exponent Lebesgue spaces (for $M(x,\xi)=|\xi|^{p(x)}$ with $1<p_{\rm min}\leq p(x)\leq p^{\rm max}<\infty$) are still reflexive. Despite the methods of analysis of problems in this setting are more advanced, they are in the same spirit.

Studies on renormalized solutions comes from DiPerna and Lions [12] investigations on the Boltzmann equation. In the elliptic setting the foundations of the branch were laid by Boccardo et. al. [8], Dall’Aglio [11] and Murat [33], providing results for operators with polynomial growth. Their generalisations to the variable exponent setting can be find in [3, 4, 42].

Investigations of nonlinear elliptic boundary value problems in non-reflexive Orlicz-Sobolev-type setting was initiated by Donaldson [13] and continued by Gossez [16, 17, 18]. For a summary of the results we refer to [35] by Mustonen and Tienari. The generalization to the case of vector Orlicz spaces with possibly anisotropic modular function, but independent of spacial variables was investigated in [21].

The existence theory for problems in this setting arising from fluids mechanics is developed from various points of view [20, gwiazda-tmna, 22, 43]. For the recent existence results for elliptic problems we refer to [1, 2, 5, 6, 14, 15, 29, 24, 25, 27, 29, 30]. In [15, 27, 30] isotropic, separable and reflexive Musielak-Orlicz spaces are employed, [5] concerns anisotropic variable exponent spaces, [14] studies separable, but not reflexive Musielak-Orlicz spaces, while [29] anisotropic, but separable and reflexive Orlicz spaces. Renormalized solutions to elliptic problems in Orlicz spaces are explored in [1, 2, 6], while in Musielak-Orlicz spaces in [24, 25].

Approximation in Musielak-Orlicz spaces

The highly challenging part of analysis in the general Musielak-Orlicz spaces is giving a relevant structural condition implying approximation properties of the space. However, we are equipped not only with the weak-* and strong topology of the gradients, but also with the intermediate one, namely - the modular topology.

In the mentioned existence results even in the case, when the growth conditions imposed on the modular function were given by a general $N$-function, besides the growth condition on $M^{*}$, also $\Delta_{2}$-condition on $M$ was assummed (which entails separability of $L_{M^{*}}$, see [43]). It results further in density of smooth functions in $L_{M}$ with respect to the weak-$*$ topology. In the case of classical Orlicz spaces, the crucial density result was provided by Gossez [18]. The improvement of this result for the vector Orlicz spaces was given in [21], while for the $x$–dependent log-Hölder continuous modular functions in [7], developed in [19, 40] and further in [41] in the case of log-Hölder continuous modular functions dependent on $x$, as well as on $t$.

Let us discuss our assumption (M). First we shall stress that it is applied only in the proof of approximation result (Theorem 2.2). When we deal with the space equipped with the approximation properties, we can simply skip (M). Namely, this is the case e.g. of the following modular functions:

• •

$M(x,|\xi|)=|\xi|^{p}+a(x)|\xi|^{q}$, where $1\leq p<q$ and function $a$ is nonnegative a.e. in $\Omega$ and $a\in L^{\infty}(\Omega)$, covering the celebrated double-phase case [10];

• •

$M(x,\xi)=M_{1}(\xi)+a(x)M_{2}(\xi)$, where $M_{1},M_{2}$ satisfy conditions $\Delta_{2}$ and $\nabla_{2}$, moreover a function $a$ is nonnegative a.e. in $\Omega$ and $a\in L^{\infty}(\Omega)$.

In the both above cases modular approximation sequence obtained in the spirit of Theorem 2.2 can be replaced by existence of a strongly converging affine combination of the weakly converging sequence (ensured in any reflexive Banach space via Mazur’s Lemma).

In the variable exponent case typical assumption resulting in approximation properties of the space is log-Hölder continuity of the exponent. In the isotropic case (when $M(x,\xi)=M(x,|\xi|)$) Lemma 3.2 shows that to get (M), it suffices to impose on $M$ continuity condition of log-Hölder-type with respect to $x$, namely for each $\xi\in{\mathbb{R}^{N}}$ and $x,y,$ such that $|x-y|<\frac{1}{2}$ we have

[TABLE]

Note that condition (6) for $M(x,\xi)=|\xi|^{p(x)}$ relates to the log-Hölder continuity condition for the variable exponent $p$, namely there exists $a>0$, such that for $x,y$ close enough and each $\xi\in{\mathbb{R}^{N}}$

[TABLE]

Indeed,

[TABLE]

There are several types of understanding generalisation of log-Hölder continuity to the case of general $x$-dependent isotropic modular functions (when $M(x,\xi)=M(x,|\xi|)$). The important issue is the interplay between types of continuity with respect to each of the variables separately. Besides our condition (6) (sufficient for (M) via Lemma 3.2), we refer to the approaches of [27, 28] and [31, 32], where the authors deal with the modular function of the form $M(x,\xi)=|\xi|\phi(x,|\xi|)$. We proceed without their doubling assumptions ($\Delta_{2}$). Since we are restricted to bounded domains, condition $\phi(x,1)\sim 1$ follows from our definition of $N$-function (Definition 3.1 ). As for the types of continuity, in [31, 32] the authors restrict themselves to the case when $\phi(x,|\xi|)\leq c\phi(y,|\xi|)$ when $|\xi|\in[1,|x-y|^{-n}].$ This condition implies (6) and consequently (M). Meanwhile in [27, 28], the proposed condition yields $\phi(x,b|\xi|)\leq\phi(y,|\xi|)$ when $\phi(y,|\xi|)\in[1,|x-y|^{-n}],$ which does not imply (6) directly. However, we shall mention that all three conditions are of the same spirit and balance types of continuity with respect to each of the variables separately.

Our approach

The challenges resulting from the lack of the growth conditions are significant and require precise handling with general $x$-dependent and anisotropic $N$-functions. The space we deal with is, in general, neither separable, nor reflexive. Resigning from imposing $\Delta_{2}$-condition on the conjugate of the modular function $M$ complicates understanding of the dual pairing. As a further consequence of relaxing growth condition, we cannot use classical results, such as the Sobolev embeddings or the Rellich-Kondrachov compact embeddings. We extend the main goal of [19], where the authors deal with bounded data. Lack of precise control on the growth of the  leading part of the operator, together with the low integrability of the right-hand side results in noticeable difficulties in studies on convergence.

Besides the refined version of approximation result of [19] (Theorem 2.2), we prove general modular Poincaré-type inequality (Theorem 2.3). The main goal, i.e. the existence of renormalized solutions to general nonlinear elliptic equation, is given in Theorem 1.1. Our methods leading to this result are based on the scheme of [24, 25], i.e. we employ truncation arguments, the Minty-Browder monotonicity trick and the Young measures. However, unlike in the latter papers we put regularity restrictions on the modular function instead of the growth conditions.

2 Preliminaries

In this section we give only the general preliminaries concerning the setting. All necessary definitions and technical tools, as well as an introduction to the setting and general theorems are given in Appendix.

Classes of functions

Definition 2.1**.**

*Let $M$ be an $N$-function (cf. Definition 3.1).

We deal with the three Orlicz-Musielak classes of functions.*

• i)

${\cal L}_{M}(\Omega;{\mathbb{R}^{N}})$* - the generalised Orlicz-Musielak class is the set of all measurable functions $\xi:\Omega\to{\mathbb{R}^{N}}$ such that*

[TABLE]

• ii)

${L}_{M}(\Omega;{\mathbb{R}^{N}})$* - the generalised Orlicz-Musielak space is the smallest linear space containing ${\cal L}_{M}(\Omega;{\mathbb{R}^{N}})$, equipped with the Luxemburg norm*

[TABLE]

• iii)

${E}_{M}(\Omega;{\mathbb{R}^{N}})$* - the closure in $L_{M}$-norm of the set of bounded functions.*

Then

[TABLE]

the space ${E}_{M}(\Omega;{\mathbb{R}^{N}})$ is separable and $({E}_{M}(\Omega;{\mathbb{R}^{N}}))^{*}=L_{M^{*}}(\Omega;{\mathbb{R}^{N}})$, see [20, 43].

Under the so-called $\Delta_{2}$-condition (Definition 3.3) we would be equipped with stronger tools. Indeed, if $M\in\Delta_{2}$, then

[TABLE]

and $L_{M}(\Omega;{\mathbb{R}^{N}})$ is separable. When both $M,M^{*}\in\Delta_{2}$, then $L_{M}(\Omega;{\mathbb{R}^{N}})$ is separable and reflexive, see [19, 20]. We face the problem without this structure.

Remark 2.1**.**

Definition 3.1 (see points 3 and 4) implies $\lim_{|\xi|\to\infty}\inf_{x\in\Omega}\frac{M^{*}(x,\xi)}{|\xi|}=\infty$ and $\inf_{x\in\Omega}M^{*}(x,\xi)>0$ for any $\xi\neq 0$. Then, consequently, Lemma 3.1 ensures

[TABLE]

Comments on assumptions on $A$

The following consideration explains how condition (A2) settles growth and coercivity condition on the leading part of the operator.

In the standard $L^{p}$-setting it is enough to note that (A2) implies directly

[TABLE]

and $|A(x,\xi)|\cdot|\xi|\geq\widetilde{c}_{A}|A(x,\xi)|^{p^{\prime}},$ leading further to the condition

[TABLE]

In the nonstandard growth setting, considering the first counterpart of the above condition, i.e.

[TABLE]

we get the minimal growth. As for the bound from above, we define an increasing function $P:{\mathbb{R}}\cup\{0\}\to{\mathbb{R}}\cup\{0\}$ by the following formula

[TABLE]

Notice that for every $x\in\Omega$ and $\xi\in{\mathbb{R}^{N}}$ such that $|\xi|=s$ it holds $P(s)\geq M(x,\xi)$. Moreover, we have an upper bound for the growth of the operator

[TABLE]

Indeed, to prove

[TABLE]

it suffices to notice that Fechel-Young inequality (44) yields

[TABLE]

whereas on the other hand

[TABLE]

Conditions of this form are considered in classical Orlicz setting, when $M(x,\xi)=M(|\xi|)$ by e.g. [18, 35]. Note that then we can take $P(s)=M(s)$. Since (A2) implies (8) and (9), we assume particular growth and coercivity of the leading part of the operator corresponding to the modular function of the space, where the solutions are defined. Nonetheless, conditions (8) and (9) are not sufficient in our approach. Note that they do not ensure that the operator and the solution are in the proper dual spaces. Let us stress further that the consequences of (A2) are expressed by $N$-functions of general type of growth.

Main tools

The existence of solutions to the truncated problem follows directly from [19, Theorem 1.5].

Theorem 2.1** (Existence with bounded data, cf. [19]).**

Suppose $g\in L^{\infty}(\Omega)$, an $N$-function $M$ satisfies assumption (M) and function $A$ satisfies assumptions (A1)-(A3). Then there exists a weak solution to the problem

[TABLE]

Namely, there exists $u\in W_{0}^{1,1}(\Omega)$ such that $\nabla u\in L_{M}(\Omega)$ satisfies

[TABLE]

for all $\varphi\in C_{0}^{\infty}(\Omega)$. Moreover, $A(\cdot,\nabla u)\in L_{M^{∗}}(\Omega)$.

In fact, [19, Theorem 1.5] is proven under the assumption that there exists $F:\Omega\to{\mathbb{R}^{N}}$, such that $g=\mathrm{div}F$ and $F\in E_{M^{*}}(\Omega)$. Nevertheless, each bounded $g$ is of this form. Existence of such $F$ is clear, while the fact that $F\in E_{M^{*}}(\Omega)$ is a consequence of properties of the Bogovski operator, see e.g. [[39], Lemma II.2.1.1].

The following refined approximation result of [19, Theorem 2.7] being an improvement of the case from [7] is proven in Appendix.

Theorem 2.2** (Approximation theorem).**

Let $\Omega$ be a Lipschitz domain and an $N$-function $M$ satisfy condition (M). Then for any $\varphi$ such that $\varphi\in V_{0}^{M}\cap L^{\infty}(\Omega)$ there exists a sequence $\{\varphi_{\delta}\}_{\delta>0}\in C_{0}^{\infty}(\Omega)$ converging modularly to $\varphi$, i.e. such that $\nabla\varphi_{\delta}\xrightarrow[]{M}\nabla\varphi$.

The vital tool in our study is the following modular Poincaré-type inequality. The proof is also included in Appendix.

Theorem 2.3** (Modular Poincaré inequality).**

Let $m:{\mathbb{R}_{+}}\to{\mathbb{R}_{+}}$ be an arbitrary function satisfying $\Delta_{2}$-condition and $\Omega\subset{\mathbb{R}^{N}}$ be a bounded domain, then there exist $c=c(\Omega,N,m)>0$ such that for every $g\in W^{1,1}(\Omega)$, such that $\int_{\Omega}m(|\nabla g|)dx<\infty$, we have

[TABLE]

3 The main proof

Proof of Theorem 1.1.

The proof is divided into several steps.

Step 1. Truncated problem. Existence to a truncated problem

[TABLE]

for $s>0$ is a direct consequence of Theorem 2.1 with $g=T_{s}(f)$ (truncation $T_{s}$ comes from (5)).

Step 2. A priori estimates. In order to obtain uniform integrability of sequences $\{A(x,\nabla T_{k}(u_{s}))\}_{s>0}$ and $\{\nabla T_{k}(u_{s})\}_{s>0}$ we need to obtain the following a priori estimates.

For $u_{s}$ being a weak solution to (10), $s>0$ and $f\in L^{1}(\Omega)$, we have the following estimates for any $k>0$

[TABLE]

where the constant $c$ depends only on the growth condition (A2).

Indeed, considering $(T_{k}(u_{s}))_{\delta}$ – a sequence approximating $T_{k}(u_{s})$ as in Theorem 2.2, we get

[TABLE]

We observe that due to Assumption (A2) we have

[TABLE]

Estimates (11) and (12) are direct consequences of the above one. Then, according to Lemma 3.3, we reach the goal of this step.

Step 3. Controlled radiation. The proof of this step is a modification of [24, Lemma 5.1, Corollary 5.2]. We consider the $N$-function ${\underline{m}}:{\mathbb{R}}_{+}\cup\{0\}\to{\mathbb{R}}$ defined as follows. Let

[TABLE]

Then, let ${\underline{m}}$ be a solution to the differential equation

[TABLE]

with the initial condition ${\underline{m}}(0)=0=m_{*}(0)$ and a certain $\alpha>1$. Note that ${\underline{m}}^{\prime}(s)\leq m^{\prime}_{*}(s)$ for every $s$, so ${\underline{m}}(s)\leq m_{*}(s)$. Due to Lemma 3.3 also $m_{*}(s)\leq\inf_{x\in\Omega,\ |\xi|=s}M(x,\xi)$ for every $s$. Thus

[TABLE]

Moreover, by [38, Chapter II.2.3, Theorem 3, point 1. (ii)] ${\underline{m}}$ satisfies $\Delta_{2}$-condition (cf. (54) without dependence on $x$).

Proposition 3.1**.**

Suppose $u_{s}$ is a weak solution to (10), $s>0$ and $f\in L^{1}(\Omega)$. Then there exist $c>0$ and $\gamma:{\mathbb{R}}_{+}\to{\mathbb{R}}_{+}$, such that for every $l>0$

[TABLE]

and $\gamma$ is independent of $l,s$ and $\lim_{r\to 0}\gamma(r)=0$.

Proof.

Note that for ${\underline{m}}$ given by (13) we have

[TABLE]

Moreover, for $l>0$ we have

[TABLE]

In the above estimates we apply (respectively) the Chebyshev inequality, the Poincaré inequality (Theorem 2.3), a priori estimate (11) and the facts that $f\in L^{1}(\Omega)$ and that ${\underline{m}}$ is an $N$-function (cf. Definition 3.1). Thus, there exists $\gamma:{\mathbb{R}}_{+}\to{\mathbb{R}}_{+}$ independent of $l,s$, for which $\lim_{r\to\infty}\gamma(r)=0$. Moreover, $\int_{E}|f|\,dx\leq\gamma(|E|).$ In particular,

[TABLE]

As for the second assertion let us define $\psi_{l}:{\mathbb{R}}\to{\mathbb{R}}$ by

[TABLE]

and consider $(\psi_{l}(u_{s}))_{\delta}$ – a sequence approximating $\psi_{l}(u_{s})$ as in Theorem 2.2. Using $(1-\psi_{l}(u_{s}))_{\delta}$ as a test function in (10) we get

[TABLE]

We notice that the meaning of truncations and the form of $\psi_{l}$, together with (15) implies

[TABLE]

which was the aim. ∎

Step 4. Convergence of truncations

Proposition 3.2**.**

Suppose an $N$-function $M$ satisfies assumption (M) and function $A$ satisfies assumptions (A1)-(A3). For $s>0$ and $f\in L^{1}(\Omega)$ let $u_{s}$ be a weak solution to (10). Then there exists a measurable function $u:\Omega\to{\mathbb{R}}$, such that $T_{k}(u)\in V_{0}^{M}$, being a limit of some subsequence of $\{u_{s}\}_{s}$ in the following sense

[TABLE]

and for each $k\in{\mathbb{N}}$ and $s\to\infty$

[TABLE]

Proof.

The proven a priori estimate (11)

[TABLE]

implies that for each $k$ the sequence $(T_{k}(u_{s}))_{s=1}^{\infty}$ is bounded in $W^{1,1}_{0}(\Omega)$. Hence, there exists a function $u$ such that

[TABLE]

in particular implying (20) and (21). Furthermore, the Lebesgue Monotone Convergence Theorem implies

[TABLE]

and up to a subsequence we have (17), i.e.

[TABLE]

Since $\Omega$ is bounded, for fixed $k\in{\mathbb{N}}$ convergence in (19) results from uniform integrability in $L^{p}(\Omega)$ of bounded functions $T_{k}(u_{s})$ combined with the Vitali Convergence Theorem (Theorem 3.2). Meanwhile, the Dominated Convergence Theorem (due to (15)) gives (18).

On the other hand, if for every $k$ we denote

[TABLE]

then it follows from (12) that there exists ${\cal{A}}_{k}\in L_{M^{*}}(\Omega;{\mathbb{R}^{N}})$ such that

[TABLE]

Our aim is now to show that in (23)

[TABLE]

We take approximating sequence of smooth functions $\nabla(T_{k}(u))_{\delta}\xrightarrow[\delta\to 0]{M}\nabla T_{k}(u)$ (cf. Theorem 2.2) and show that

[TABLE]

Testing (10) by $\varphi=\psi_{l}(u_{s})(T_{k}(u_{s})-(T_{k}(u))_{\delta}),$ where $\psi_{l}$ is given by (16), we get

[TABLE]

We observe that the right-hand side of (26) tends to zero, i.e.

[TABLE]

Indeed, the convergence a.e. is ensured by (17) and to apply the Lebesgue Dominated Convergence Theorem we note

[TABLE]

The last expression is convergent due to Lemma 3.5.

Let us now concentrate on the left-hand side of (26):

[TABLE]

where due to (14) we have

[TABLE]

Moreover,

[TABLE]

meanwhile the convergence of $II_{2}$ results from Lemma 3.5.

Then passing to the limit in (26) we obtain

[TABLE]

Then (27) is equivalent to (25).

Before we apply monotonicity trick, we need to show that

[TABLE]

Taking into account (25), the equality (28) will be proven when the following expression is shown to tend to [math] (still $k\leq l$)

[TABLE]

We prove that

[TABLE]

For this we will use Lemma 3.5 with

[TABLE]

The convergence $w^{s}\xrightharpoonup{}w$ is a consequence of (23). Let $v^{s}=\mathds{1}_{\{k<|u_{s}|\}}\psi_{l}(u_{s})$ and $v^{s}_{\varepsilon}\in C(\Omega)\cap L^{\infty}(\Omega)$ with ${\varepsilon}\geq 0$ be given by

[TABLE]

Notice that for $s\to\infty$ and every ${\varepsilon}>0$, due to continuity of $v_{\varepsilon}^{s}$, we have

[TABLE]

Furthermore, for every $s$ we have

[TABLE]

Since $v_{\varepsilon}\in L^{\infty}(\Omega)$, Lemma 3.5 yields $\int_{\Omega}w^{s}v^{s}_{\varepsilon}\,dx\to\int_{\Omega}wv_{\varepsilon}\,dx,$ that is

[TABLE]

The Lebesgue Monotone Convergence Theorem implies

[TABLE]

Thus (31) together with (32) give

[TABLE]

and we get (30).

Our aim now is to prove

[TABLE]

Recall that $\nabla(T_{k}(u))_{\delta}\xrightarrow{M}\nabla T_{k}(u)$. Therefore by Definition 3.4 ii), the sequence $\{M(x,\nabla(T_{k}(u))_{\delta}/\lambda)\}_{\delta}$ is uniformly bounded in $L^{1}(\Omega;{\mathbb{R}^{N}})$ for some $\lambda$ and consequently, by Lemma 3.3 $\{\nabla(T_{k}(u))_{\delta}\}_{\delta}$ is uniformly integrable. Hence the Vitali Convergence Theorem (Theorem 3.2) gives

[TABLE]

which is equal to zero, because $T_{k}(u)|\mathds{1}_{\{k<|u|\}}=0$. Thus (33) and (28) hold.

We observe that we can remove $\psi_{l}(u_{s})$ from (28). Indeed, notice that for $l\geq k$ due to Lemma 3.1 we have

[TABLE]

Therefore, (28) is equivalent to

[TABLE]

Now we apply the Minty-Browder monotonicity trick. Since (23), then for each $\delta$

[TABLE]

Then (34) together with (35) imply

[TABLE]

where the last equality is obtained analogically as (33).

Monotonicity of $A$ results in

[TABLE]

for any $\eta\in{\mathbb{R}^{N}}$. Taking upper limit with ${s\to\infty}$ above (due to (36), (23), and (21)) we get

[TABLE]

Note that it is equivalent to

[TABLE]

Let us define

[TABLE]

Then, in (37) we choose

[TABLE]

where $0<j<i$, $h\in{\mathbb{R}_{+}}$ and $z\in L^{\infty}(\Omega;{\mathbb{R}^{N}})$, to get

[TABLE]

Notice that it is equivalent to

[TABLE]

The first and the second expression above tend to zero when $i\to\infty.$ Indeed, since ${\cal A}_{k},A(x,0)\in L_{M^{*}}(\Omega;{\mathbb{R}^{N}})$ and $\nabla T_{k}(u)\in L_{M}(\Omega;{\mathbb{R}^{N}})$, the Hölder inequality (45) gives boudedness of integrands in $L^{1}(\Omega)$. Then we take into account shrinking domain of integration to get the desired convergence to [math]. In particular, we can erase these expressions in (39) and divide the remaining expression by $h>0$, to obtain

[TABLE]

Note that

[TABLE]

Moreover, as $A(x,\nabla T_{k}(u)+hz)$ is bounded on $\Omega_{j}$, Lemma 3.1 results in

[TABLE]

The right-hand side is bounded, because $(\nabla T_{k}(u)+hz)_{h}$ is uniformly bounded in $L^{\infty}(\Omega_{j};{\mathbb{R}^{N}})\subset L_{M}(\Omega;{\mathbb{R}^{N}})$ (cf. (38) and (7)). Hence, Lemma 3.3 gives uniform integrability of $\left(A(x,\nabla T_{k}(u)+hz)\right)_{h}$. When we notice that $|\Omega_{j}|<\infty$, we can apply the Vitali Convergence Theorem (Theorem 3.2) to get

[TABLE]

Thus

[TABLE]

Consequently,

[TABLE]

for any $z\in L^{\infty}(\Omega;{\mathbb{R}^{N}})$. Let us take

[TABLE]

We obtain

[TABLE]

hence

[TABLE]

Since $j$ is arbitrary, we have the equality a.e. in $\Omega$ and (24) is satisfied. ∎

Step 5. Renormalized solutions.

We aim at proving that $u$ is a renormalized solution (see Introduction). At first we observe that $u$ satisfies (R1) and concentrate on (R2).

Since $T_{k}(u)\in V_{0}^{M}\cap L^{\infty}(\Omega)$, Theorem 2.2 ensures that there exists a sequence $\{u_{r}\}_{r}\subset C_{0}^{\infty}(\Omega)$ indexed with $r\to\infty$, such that

[TABLE]

where $h\in C_{c}^{1}({\mathbb{R}})$ is arbitrary.

We test (10) by $\psi_{l}(u_{s})h(u_{r})\phi$ with $\phi\in W^{1,\infty}_{0}(\Omega)$ and get

[TABLE]

We notice at first that due to the Lebesgue Dominated Convergence Theorem it holds that

[TABLE]

Meanwhile on the left-hand side

[TABLE]

where

[TABLE]

due to (14). As for $L^{2}_{s,r,l}$ we notice that when $s\to\infty$, up to a subsequence,

[TABLE]

Indeed, a priori estimate (12) combined with Lemma 3.3 give uniform integrability. Then, taking into account weak-* convergence (22), the Dunford-Pettis Theorem (Theorem 3.3) ensures weak $L^{1}$-convergence up to a subsequence.

Moreover, note that

[TABLE]

and for $s\to\infty$

[TABLE]

The sequence $\{A(x,\nabla u_{s})\psi_{l}(u_{s})\nabla[h(u_{r})\phi]\}_{s}$ is uniformly integrable. Due to the consequence of Chacon’s Biting Lemma, Theorem 3.1, we notice that

[TABLE]

Since $\mathrm{supp}\,h(u)\subset[-m,m]$ for some $m\in{\mathbb{N}}$ and we can consider only $l>m+1$. Then

[TABLE]

and our solution $u$ satisfies condition (R2).

Let us consider radiation control condition (R3), i.e.

[TABLE]

We follow the ideas of [26] involving the Chacon Biting Lemma and the Young measure approach to show that for $s\to\infty$ it holds that

[TABLE]

First we observe that the sequence $\{[{{\cal A}_{s,l+1}}-A(x,\nabla T_{l+1}(u))]\cdot[\nabla T_{l+1}(u_{s})-\nabla T_{l+1}(u)]\}_{s}$ is uniformly bounded in $L^{1}(\Omega)$. Indeed,

[TABLE]

where $IV_{1}$ is uniformly bounded due to (14) and $IV_{4}$ is independent of $s$. As for $IV_{2}$ we note

[TABLE]

where we applied (34), (22), and then (36). Moreover, in the case of $IV_{3}$ the Fenchel-Young inequality and (11) gives boundedness.

Then monotonicity of $A(x,\cdot)$ and Theorem 3.1 give, up to a subsequence, convergence

[TABLE]

where $\nu_{x}$ denotes the Young measure generated by the sequence $\{\nabla T_{l+1}(u_{s})\}_{s}$.

Since $\nabla T_{l+1}(u_{s})\xrightharpoonup{}\nabla T_{l+1}(u)$ in $L^{1}(\Omega)$, we have $\int_{\mathbb{R}^{N}}\lambda\,d\nu_{x}(\lambda)=\nabla T_{l+1}(u)$ for a.e. $x\in\Omega$. Then

[TABLE]

and the limit in (41) is equal for a.e. $x\in\Omega$ to

[TABLE]

Uniform boundedness of the sequence $\{{{\cal A}_{s,l+1}}\nabla T_{l+1}(u_{s})\}_{s}$ in $L^{1}(\Omega)$ (cf. (14)) enables us to apply once again Theorem 3.1 to obtain

[TABLE]

Moreover, assumption (A2) implies ${{\cal A}_{s,l+1}}\nabla T_{l+1}(u_{s})\geq 0$. Therefore, due to (42) and (41), we have

[TABLE]

Taking into account that in (36) we can put ${\cal A}_{k}=A(x,\nabla T_{l+1}(u))=\int_{\mathbb{R}^{N}}A(x,\lambda)\,d\nu_{x}(\lambda)$, the above expression implies

[TABLE]

When we apply it, together with (42), the limit in (41) is non-positive. Hence,

[TABLE]

Observe further that $A(x,\nabla T_{l+1}(u))\in L_{M^{*}}(\Omega;{\mathbb{R}^{N}})$ and we can choose ascending family of sets $E^{l+1}_{j}$, such that $|E^{l+1}_{j}|\to 0$ for $j\to\infty$ and $A(x,\nabla T_{l+1}(u))\in L^{\infty}(\Omega\setminus E^{l+1}_{j}).$ Then, since $\nabla T_{l+1}(u_{s})\xrightharpoonup{}\nabla T_{l+1}(u)$, we get

[TABLE]

and similarly we conclude

[TABLE]

Summing it up we get

[TABLE]

Recall that Theorem 3.1 together with (36) and (22) results in (40).

We turn back to prove (R3). Note that $\nabla u_{s}=0$ a.e. in $\{x\in\Omega:|u_{s}|\in\{l,l+1\}\}$. Then (14) implies

[TABLE]

For $g_{l}:{\mathbb{R}}\to{\mathbb{R}}$ defined by

[TABLE]

we have

[TABLE]

Let us remind that we know that $u_{s}\to u$ a.e. in $\Omega$ (cf. (17)) and $|\{x:|u_{s}|>l\}|\leq\gamma\left(l/{\underline{m}}(l)\right)$ (cf. (18)). Moreover, we have weak convergence (40), $A(x,\nabla T_{l+2}(u_{s}))\cdot\nabla T_{l+2}(u_{s})>0$ and function $g_{l}$ is continuous and bounded. Thus, we infer that we can estimate the limit of the right-hand side of (43) in the following way

[TABLE]

where the last equality comes from (14).

Hence, our solution $u$ satisfies condition (R3) and is a renormalized solution. ∎

Appendix A

Definition 3.1** ($N$-function).**

Suppose $\Omega\subset{\mathbb{R}^{N}}$ is an open bounded set. A function $M:\Omega\times{\mathbb{R}^{N}}\to{\mathbb{R}}$ is called an $N$-function if it satisfies the following conditions:

$M$* is a Carathéodory function (i.e. measurable with respect to $x$ and continuous with respect to the last variable), such that $M(x,\xi)=0$ if and only if $\xi=0$; and $M(x,\xi)=M(x,-\xi)$ a.e. in $\Omega$,* 2. 2.

$M(x,\xi)$* is a convex function with respect to $\xi$,* 3. 3.

$\lim_{|\xi|\to 0}\mathrm{ess\,sup}_{x\in\Omega}\frac{M(x,\xi)}{|\xi|}=0$, 4. 4.

$\lim_{|\xi|\to\infty}\mathrm{ess\,inf}_{x\in\Omega}\frac{M(x,\xi)}{|\xi|}=\infty$.

Definition 3.2** (Complementary function).**

The complementary function $M^{*}$ to a function $M:\Omega\times{\mathbb{R}^{N}}\to{\mathbb{R}}$ is defined by

[TABLE]

Remark 3.1**.**

If $f(x,\xi)\leq g(x,\xi)$, then $g^{*}(x,\xi)\leq f^{*}(x,\xi)$.

Remark 3.2**.**

If $M$ is an $N$-function and $M^{*}$ its complementary, we have

• •

the Fenchel-Young inequality

[TABLE]

• •

the generalised Hölder’s inequality

[TABLE]

Lemma 3.1**.**

Suppose $M$ and $A$ are such that (A2) is satisfied, then

[TABLE]

Proof.

Since $M^{*}$ is convex, $M^{*}(x,0)=0$ and $c_{A}\in(0,1]$, we notice that

[TABLE]

Taking this into account together with (A2) and (44) we have

[TABLE]

We can ignore $M(x,\eta)>0$ on the left-hand side above, rearrange the remaining terms and integrate both sides over $\Omega$ (cf. (7)) to get the claim. ∎

Remark 3.3**.**

For any function $f:{\mathbb{R}}^{M}\to{\mathbb{R}}$ the second conjugate function $f^{**}$ is convex and $f^{**}(x)\leq f(x)$. In fact, $f^{**}$ is a convex envelope of $f$, namely it is the biggest convex function smaller or equal to $f$.

Lemma 3.2**.**

Suppose a cube ${Q_{j}^{\delta}}$ is an arbitrary one defined in (M) with $\delta_{0}=1/(8\sqrt{N})$ and function $M:{\mathbb{R}^{N}}\times[0,\infty)\rightarrow[0,\infty)$ is log-Hölder continuous, that is there exist constants $a_{1}>0$ and $b_{1}\geq 1$, such that for all $x,y\in\Omega$ with $|x-y|\leq\frac{1}{2}$ and all $\xi\in{\mathbb{R}^{N}}$ we have (6). Let us consider function ${M_{j}^{\delta}}$ given by (4) and its greatest convex minorant $({M_{j}^{\delta}})^{**}$. Then there exist constants $a,c>0$, such that (3) is satisfied.

Proof. cf. [9].

First, we fix an arbitrary $y\in Q^{\delta}_{j}$ and note that

[TABLE]

We estimate separately both quotients on the right hand side of the latter equality. By continuity of $M$ we find $\bar{y}\in\widetilde{Q}^{\delta}_{j}$ such that $M^{\delta}_{j}(\xi)=M(\bar{y},\xi)$. Then using condition (6) and the fact that $|y-\bar{y}|\leq 3\delta\sqrt{d}<\frac{1}{2}$ we get

[TABLE]

In order to estimate the second quotient in (46) we observe first that if $\xi\in[0,\infty)$ is such that $M^{\delta}_{j}(\xi)=(M^{\delta}_{j})^{**}(\xi)$ then the statement is obvious. Therefore we assume that $M^{\delta}_{j}(\xi_{0})>(M^{\delta}_{j})^{**}(\xi_{0})$ at some $\xi_{0}$. Due to continuity of $M^{\delta}_{j}$ and $(M^{\delta}_{j})^{**}$ there is a neighborhood $U$ of $\xi_{0}$ such that $M^{\delta}_{j}>(M^{\delta}_{j})^{**}$ on $U$. Consequently, $(M^{\delta}_{j})^{**}$ is affine on $U$. Moreover, Definition 3.1 implies that $m_{1}\leq M^{\delta}_{j}\leq m_{2}$, where $m_{1}$ and $m_{2}$ are convex. Therefore there are $\xi_{1},\xi_{2}$ such that $U\subset(\xi_{1},\xi_{2})$, $M^{\delta}_{j}>(M^{\delta}_{j})^{**}$ on $(\xi_{1},\xi_{2})$, $(M^{\delta}_{j})^{**}(\xi_{i})=M^{\delta}_{j}(\xi_{i})$, $i=1,2$ and $(M^{\delta}_{j})^{**}$ is an affine function on $[\xi_{1},\xi_{2}]$, i.e.

[TABLE]

We note that we consider $\xi_{1}>0$, because it follows that $0=M^{\delta}_{j}(0)=(M^{\delta}_{j})^{**}(0)$. Now, thanks to the continuity of $M$ we find $y_{i}\in\widetilde{Q}^{\delta}_{j}$ such that $M^{\delta}_{j}(\xi_{i})=M(y_{i},\xi_{i})$, $i=1,2$. Consequently, it follows from (48) that

[TABLE]

Denoting $\tilde{\xi}=t\xi_{1}+(1-t)\xi_{2}$ we get

[TABLE]

Next, we observe that the definition of $M^{\delta}_{j}$ implies $M(y_{1},\xi_{1})=M^{\delta}_{j}(\xi_{1})\leq M(y_{2},\xi_{1})$. We can assume without loss of generality that

[TABLE]

because for $M(y_{1},\xi_{1})=M(y_{2},\xi_{1})$ inequality (50) implies $M^{\delta}_{j}\leq(M^{\delta}_{j})^{**}$ on $[\xi_{1},\xi_{2}]$. Since we have always $M^{\delta}_{j}\geq(M^{\delta}_{j})^{**}$ we arrive at $M^{\delta}_{j}=(M^{\delta}_{j})^{**}$ on $[\xi_{1},\xi_{2}]$.

Let us consider a function $h:[0,1]\rightarrow{\mathbb{R}}$ defined by

[TABLE]

Then we compute

[TABLE]

Obviously, we have $h^{\prime}>0$ on $(0,1)$ due to (51). Therefore the maximum of $h$ is attained at $t=1$, which implies

[TABLE]

Next, we apply condition (6) and $\xi_{1}\leq\tilde{\xi}$ to infer

[TABLE]

since $y_{1},y_{2}\in\tilde{Q}^{\delta}_{j}$ implies $|y_{1}-y_{2}|\leq 4\delta\sqrt{N}<\frac{1}{2}$. Combining (46) with (47) and (53) yields

[TABLE]

which is the desired conclusion. ∎

Definition 3.3** ($\Delta_{2}$-condition).**

We say that an $N$-function $M:\Omega\times{\mathbb{R}^{N}}\to{\mathbb{R}}$ satisfies $\Delta_{2}$ condition if for a.e. $x\in\Omega$, there exists a constant $c>0$ and nonnegative integrable function $h:\Omega\to{\mathbb{R}}$ such that

[TABLE]

Appendix B

We have two equivalent definitions of modular convergence.

Definition 3.4** (Modular convergence).**

We say that a sequence $\{\xi_{i}\}_{i=1}^{\infty}$ converges modularly to $\xi$ in $L_{M}(\Omega;{\mathbb{R}^{N}})$ (and denote it by $\xi_{i}\xrightarrow[i\to\infty]{M}\xi$), if

• i)

there exists $\lambda>0$ such that

[TABLE]

equivalently

• ii)

there exists $\lambda>0$ such that

[TABLE]

Definition 3.5** (Biting convergence).**

Let $f_{n},f\in L^{1}(\Omega)$ for every $n\in{\mathbb{N}}$. We say that a sequence $\{f_{n}\}_{n=1}^{\infty}$ converges in the sense of biting to $f$ in $L^{1}(\Omega)$ (and denote it by $f_{n}\xrightarrow[]{b}f$), if there exists a sequence of measurable $E_{k}$ – subsets of $\Omega$, such that $\lim_{k\to\infty}|E_{k}|=0$, such that for every $k$ we have $f_{n}\to f$ in $L^{1}(\Omega\setminus E_{k})$.

Definition 3.6** (Uniform integrability).**

We call a sequence $\{f_{n}\}_{n=1}^{\infty}$ of measurable functions $f_{n}:\Omega\to{\mathbb{R}^{N}}$ uniformly integrable if

[TABLE]

equivalently (cf. [23]) if

[TABLE]

where we denote the positive part of function $f$ by $(f(x))_{+}:=\max\{f(x),0\}$.

We use the following results.

Lemma 3.3** (Modular-uniform integrability, [22]).**

Let $M$ be an $N$-function and $\{f_{n}\}_{n=1}^{\infty}$ be a sequence of measurable functions such that $f_{n}:\Omega\to{\mathbb{R}^{N}}$ and $\sup_{n\in{\mathbb{N}}}\int_{\Omega}M(x,f_{n}(x))dx<\infty$. Then the sequence $\{f_{n}\}_{n=1}^{\infty}$ is uniformly integrable.

Lemma 3.4** (Density of simple functions, [34]).**

Suppose (2). Then the set of simple functions integrable on $\Omega$ is dense in $L_{M}(\Omega)$ with respect to the modular topology.

The above result can be obtained by the method of the proof of [34, Theorem 7.6].

We need the following consequence of the Chacon Biting Lemma, [37, Lemma 6.9].

Theorem 3.1**.**

Let $f_{n}\in L^{1}(\Omega)$ for every $n\in{\mathbb{N}}$, $f_{n}(x)\geq 0$ for every $n\in{\mathbb{N}}$ and a.e. $x$ in $\Omega$. Moreover, suppose $f_{n}\xrightarrow[]{b}f$ (cf. Definition 3.5) and $\limsup_{n\to\infty}\int_{\Omega}f_{n}dx\leq\int_{\Omega}fdx.$ Then $f_{n}\xrightharpoonup{}f$ in $L^{1}(\Omega)$ for $n\to\infty$.

Theorem 3.2** (The Vitali Convergence Theorem).**

Let $(X,\mu)$ be a positive measure space, $\mu(X)<\infty$, and $1\leq p<\infty$. If $\{f_{n}\}$ is uniformly integrable in $L^{p}_{\mu}$, $f_{n}(x)\to f(x)$ in measure and $|f(x)|<\infty$ a.e. in $X$, then $f\in{L}^{p}_{\mu}(X)$ and $f_{n}(x)\to f(x)$ in ${L}^{p}_{\mu}(X)$.

Theorem 3.3** (The Dunford-Pettis Theorem).**

A sequence $\{f_{n}\}_{n}$ is uniformly integrable in $L^{1}(\Omega)$ if and only if it is relatively compact in the weak topology.

Lemma 3.5**.**

Suppose $w_{n}\xrightharpoonup[n\to\infty]{}w$ in $L^{1}(\Omega)$, $v_{n},v\in L^{\infty}(\Omega)$, and $v_{n}\xrightarrow[n\to\infty]{a.e.}v$. Then

[TABLE]

Appendix C

Proof of Theorem 2.2.

The proof is divided into four steps. We start with the case of star-shaped domain and then, in the fourth step, we turn to any Lipschitz domain.

Step 1. Let us assume, that $\Omega$ is a star-shape domain with respect to the ball $B(0,r)$ (i.e. with respect to any point of this ball). For $0<\delta<r/4$, we set $\kappa_{\delta}=1-\frac{2\delta}{r}$. It holds that

[TABLE]

For a measurable function $\xi:{\mathbb{R}^{N}}\to{\mathbb{R}^{N}}$ with $\mathrm{supp}\,\xi\subset\Omega$, we define

[TABLE]

where ${\varrho}_{\delta}(x)={\varrho}(x/\delta)/\delta^{N}$ is a standard regularizing kernel on ${\mathbb{R}^{N}}$ (i.e. ${\varrho}\in C^{\infty}({\mathbb{R}^{N}})$, $\mathrm{supp}\,{\varrho}\subset\subset B(0,1)$ and ${\int_{\Omega}}{\varrho}(x)dx=1$, ${\varrho}(x)={\varrho}(-x)\geq 0$). Let us notice that $\xi_{\delta}\in C_{0}^{\infty}({\mathbb{R}^{N}};{\mathbb{R}^{N}})$.

Step 2. We show that the family of operators $(\xi_{\delta})_{\delta}$ is uniformly bounded from $L_{M}(\Omega;{\mathbb{R}^{N}})$ to $L_{M}(\Omega;{\mathbb{R}^{N}})$. Without loss of generality we assume

[TABLE]

We have to show that

[TABLE]

for every suffciently small $\delta$.

We consider $M_{j}^{\delta}(\xi)$ given by (4) and ${({M_{j}^{\delta}}(\xi))^{**}}$, see Remark 3.3. Since $M(x,\xi_{\delta}(x))=0$ whenever $\xi_{\delta}(x)=0$, we have

[TABLE]

Our aim is to show now the following uniform bound

[TABLE]

for sufficiently small $\delta>0$, $x\in{Q_{j}^{\delta}}\cap\Omega$ with $c$ independent of $\delta,x$ and $j$. Let us fix an arbitrary cube and take $x\in{Q_{j}^{\delta}}$. For sufficiently small $\delta$ (i.e. $\delta<\delta_{1}:=\min\{{r}/{4},\delta_{0}\}$), due to (3), we obtain

[TABLE]

To estimate the right–hand side of (61) we consider (56). Denote

[TABLE]

Note that for any $x,y\in\Omega$ and each $\delta>0$ we have

[TABLE]

Therefore, taking into account (57) we get

[TABLE]

Note that $(2K)^{-a/\log(b\delta)}\leq(4K)^{-a/\log(b\delta_{0})}$ and

[TABLE]

which is bounded for $\delta\in[0,\delta_{0}]$. We combine this with (61) and (62) to get

[TABLE]

Thus, we have obtained (60). Now, starting from (59), noting (60) and the fact that ${({M_{j}^{\delta}}(\xi))^{**}}$=0 if and only if $\xi=0$, we observe

[TABLE]

Note that by applying the Jensen inequality the right-hand side above can be estimated by the following quantity

[TABLE]

We applied inequality for convolution, boundedness of ${\varrho}_{\delta}$, once again the fact that ${({M_{j}^{\delta}}(\xi))^{**}}$=0 if and only if $\xi=0$. Then, by the definition of $M_{j}^{\delta}(\xi)$, i.e. (4) and properties of ${({M_{j}^{\delta}}(\xi))^{**}}$, see Remark 3.3, we realize that

[TABLE]

The last inequality above stands for computation of a sum taking into account the measure of repeating parts of cubes.

We get (58) by summing up the estimates of this step.

Step 3. Fix arbitrary $\varphi\in V_{0}^{M}$ and recall definition of the cadidate for approximating family (56). We are going to show that (still in the case of star-shape domains) it holds that

[TABLE]

Fix $\sigma$ to be specified later and recall $C$ from (58). By Lemma 3.4 and continuity of $M$ we can choose family of measurable sets $\{E_{j}\}_{j=0}^{n}$ such that $\bigcup_{j=0}^{n}E_{j}=\Omega$ and a simple vector valued function

[TABLE]

such that

[TABLE]

Then by (58) we have

[TABLE]

Convexity of $M(x,\cdot)$ implies

[TABLE]

Since we have already estimated the first and the last expression on the right-hand side above, let us concentrate on the second one. The Jensen inequality and then the Fubini theorem lead to

[TABLE]

Using the continuity of the shift operator in $L^{1}$ we observe that poinwisely

[TABLE]

Moreover, note that

[TABLE]

and the Lebesgue Dominated Convergence Theorem provides the right-hand side of (66) converges to zero as $\delta\to 0$.

To sum up, regarding to arbitrariness of $\sigma>0$ in (64) and (65), and to the convergence of the second term we get the claim.

Step 4. If $\Omega$ is a bounded Lipschitz domain in ${\mathbb{R}^{N}}$, then there exists a finite family of open sets $\{\Omega_{i}\}_{i\in I}$ and a finite family of balls $\{B^{i}\}_{i\in I}$ such that

[TABLE]

and every set $\Omega_{i}$ is star-shaped with respect to ball $B^{i}$ of radius $r_{i}$ (see e.g. [36]). Let us introduce the partition of unity $\theta_{i}$ with for $x\in\Omega$. Then one can decompose function $\varphi$ in the following way

[TABLE]

Let us notice that if $\nabla\varphi\in L_{M}(\Omega;{\mathbb{R}^{N}})$ and $\varphi\in L^{\infty}(\Omega)$, then $\nabla(\theta_{i}\varphi)=(\varphi\nabla\theta_{i}+\theta_{i}\nabla\varphi)\in L_{M}(\Omega;{\mathbb{R}^{N}})$. Therefore we can apply the previous arguments to every function $\theta_{i}\varphi$ of a support on a star-shaped domain $\Omega_{i}\subset\Omega$.

∎

Proof of Theorem 2.3.

The proof consist of three steps starting with the case of smooth and compactly supported functions on small cube, then turning to the Orlicz class and concluding the claim on arbitrary bounded set.

Step 1. We start the proof for $u\in C_{0}^{\infty}(\Omega)$ with $\mathrm{supp}u\subset\subset[-\frac{1}{4},\frac{1}{4}]^{N}$. Let $u$ be extended by [math] outside $\Omega$ and ${\omega_{N}}=(1,\dots,1)\in{\mathbb{R}^{N}}.$ Note that

[TABLE]

and so

[TABLE]

Then we realize that for the constant $c=\sqrt{N}/2$ we have

[TABLE]

Applying $m$, which is increasing, to both sides above and the Jensen inequality (note that our interval with the Lebesgue measure is a probability space) we get

[TABLE]

Integrating over $\Omega$ and changing the order of integration we obtain

[TABLE]

Since $m\in\Delta_{2}$, we apply (54) (with constant $c_{m,\Delta_{2}}$ and no $x$-dependence) $k$ times with the smallest $k$, such that $c(\Omega,N)<2^{k}$. Then, due to monotonicity of $m$, we get

[TABLE]

Step 2. Let us consider now an open set $\widetilde{\Omega}$, such that $\overline{\Omega}\subset\widetilde{\Omega}\subset[-\frac{1}{4},\frac{1}{4}]^{N}$. Step 1. provides that for $u\in C_{0}^{\infty}(\widetilde{\Omega})$ we have

[TABLE]

Now, we aim at showing that for each $u\in V_{0}^{m}$ the inequality also holds. Of course, each such $u$ can be regularised by convolution with a standard mollifier $\varrho_{\frac{1}{n}}$

[TABLE]

where $\frac{1}{n}<\frac{1}{2}{\rm dist}(\partial\widetilde{\Omega},\Omega)$. Such $u_{n}$ is smooth and compactly supported in $\widetilde{\Omega}$, so we have (67) for  $u_{n}$. Passing to the limit with $n\to\infty$ gives $u_{n}\to u$ and $\nabla u_{n}\to\nabla u$ a.e. in ${\mathbb{R}^{N}}$. Then continuity of $m$ gives

[TABLE]

To get the strong convergence in $L^{1}(\Omega)$ of the sequence, we are going to apply the Vitali Convergence Theorem (Theorem 3.2). It suffices to show uniform integrability of the sequence via condition (55). Function $u\in W^{1,1}(\Omega)$, so $\nabla u_{n}=\varrho_{\frac{1}{n}}*\nabla u$. The Jensen inequality implies

[TABLE]

Observe that $t\mapsto|m(t)-1/\sqrt{\delta}|_{+}$ is a convex function and the Jensen inequality implies

[TABLE]

Moreover, $m(|\nabla u|)\in L^{1}(\widetilde{\Omega})$, so for every ${\varepsilon}>0$ there exists $\delta>0$, such the right-hand side above is smaller than ${\varepsilon}$, i.e. condition (55) is satisfied and we get uniform integrability of $\{m(|\nabla u_{n}|)\}_{n}$. From (67) we notice that $m(|u|)\in L^{1}(\widetilde{\Omega})$ and due to the same arguments the sequence $\{m(|u_{n}|)\}_{n}$ is uniformly integrable.

Step 3. Suppose that $\Omega$ is arbitrary bounded set containing [math]. It is contained in the cube of the edge $D={\rm diam}\Omega$. Then $\widetilde{u}(x)=u\left(4Dx\right)$ has ${\rm supp}\,\widetilde{u}\subset\Omega_{1}\subset\left[-\frac{1}{4},\frac{1}{4}\right]^{N}.$ We have

[TABLE]

Moreover, we estimate the right-hand side as in Step 1 in order to put a constant outside the integral and the claim follows for such $\Omega$. To obtain it on an arbitrary domain we need only to observe that the Lebesgue measure is translation-invariant. ∎