Anisotropic Orlicz-Sobolev spaces of vector valued functions and Lagrange equations

TL;DR

This paper investigates properties of anisotropic Orlicz-Sobolev spaces for vector functions, establishing a variational framework for Lagrangian systems with conditions ensuring well-defined and differentiable functionals.

Contribution

It introduces a variational setting for Lagrangian systems within anisotropic Orlicz-Sobolev spaces and provides conditions for the functional's proper mathematical behavior.

Findings

Properties of anisotropic Orlicz and Orlicz-Sobolev spaces are characterized.

A variational framework for certain Lagrangian systems is developed.

Conditions for the functional to be well-defined and differentiable are established.

Abstract

In this paper we study some properties of anisotropic Orlicz and anisotropic Orlicz-Sobolev spaces of vector valued functions for a special class of G-functions. We introduce a variational setting for a class of Lagrangian Systems. We give conditions which ensure that the principal part of variational functional is finitely defined and continuously differentiable on Orlicz-Sobolev space.

Equations

$$\frac{d}{dt}{L}_v(t,u(t),\dot{u}(t))=L_x(t,u(t),\dot{u}(t)),t\in (a,b)$$

$$I(u)={\int }_{I}F(t,u,\dot{u})dt$$

$$L^G(I,{\mathbb{R}}^N)=\{u:I\to {\mathbb{R}}^N:{\int }_{I}G(u)dt\le \infty \}.$$

$$\Vert u{\Vert }_{L^G}=\inf \left\{\alpha >0:{\int }_{I}G\left(\frac{u}{\alpha }\right)dt\le 1\right\}.$$

$$W^1{L}^G(I,{\mathbb{R}}^N)=\{u\in L^G(I,{\mathbb{R}}^N):\dot{u}\in L^G(I,{\mathbb{R}}^N)\}$$

$$\Vert u{\Vert }_{W^1{L}^G}=\Vert u{\Vert }_{L^G}+\Vert \dot{u}{\Vert }_{L^G}$$

$$\Vert u{\Vert }_{1,G,\Omega }=\Vert (u,Du){\Vert }_{G,\Omega }.$$

$$W_A^1(\Omega )=\{u\in \Omega \to \mathbb{R}\text{ measurable }:u,|\nabla u|\in L_A\},$$

$$\underset{|x|\to \infty }{\lim }\frac{G(x)}{|x|}=\infty ,$$

$${\Delta }_2$$

$${\nabla }_2$$

$$G(x)=\{\begin{array}{ll}0& |x|\le 1\\ |x{|}^2-1& |x|>1\end{array}$$

$$G(\alpha x)\le \alpha G(x),$$

$$\alpha G(x)\le G(\alpha x),$$

$$0<\alpha \le \beta ⟹G(\alpha x)\le G(\beta x).$$

$$⟨a,x_0⟩+b=G(x_0)\text{ and }⟨a,x⟩+b\le G(x).$$

$$G\left(\frac{1}{\mu (I)}{\int }_{I}udt\right)\le \frac{1}{\mu (I)}{\int }_{I}G(u)dt.$$

$$G(\alpha x)\le K_1(\alpha )G(x)$$

$$G(\alpha x)\le C_{\alpha }+K_1(\alpha )G(x).$$

$$G^{\ast }(y):=\underset{x\in {\mathbb{R}}^N}{\sup }\{⟨x,y⟩-G(x)\}.$$

$${\forall }_{x,y\in {\mathbb{R}}^N}⟨x,y⟩\le G(x)+G^{\ast }(y).$$

$$g(x)=\{\begin{array}{ll}0& |x|\le 1\\ \infty & |x|>1\end{array}$$

$$G^{\ast }(x,y)=\frac{1}{4}{x}^2+\frac{3}{4}(x+y){\left(\frac{x+y}{4}\right)}^{\frac{1}{3}}.$$

$$G_1\prec G_2⟺{\exists }_{M\ge 0}{\exists }_{K>0}{\forall }_{|x|\ge M}{G}_1(x)\le G_2(Kx)$$

$$G_1\prec \prec G_2⟺{\forall }_{\alpha >0}\underset{|x|\to \infty }{\lim }\frac{G_2(\alpha x)}{G_1(x)}=\infty .$$

$$G_1\prec G_2\Rightarrow G_2^{\ast }\prec G_1^{\ast }.$$

$$|x{|}^p\prec G\prec |x{|}^q.$$

$$G(x)\le C{K}_1{M}_1^{-q}|x{|}^q,q={\log }_2(K_1).$$

$$G(x)\ge C(2K_2{)}^{-q}|x{|}^q,p=1+\frac{1}{{\log }_2(K_2)}$$

$$L^G(I,{\mathbb{R}}^n)=\left\{u:I\to {\mathbb{R}}^n:u\text{ - measurable }{\int }_{I}G\left(u\right)dt<\infty \right\}.$$

$$\Vert u{\Vert }_{L^G}=\inf \left\{\alpha >0:{\int }_{I}G\left(\frac{u}{\alpha }\right)dt\le 1\right\}.$$

Full text

1. Introduction

In this paper we make some preliminary steps for variational analysis in anisotropic Orlicz-Sobolev spaces of vector valued functions. We consider the Euler-Lagrange equation

[TABLE]

where Lagrangian is of the form $L(t,x,v)=F(t,x,v)+V(t,x)$.

If $F(v)=\frac{1}{2}|v|^{2}$ then the equation (1) reduces to $\ddot{u}(t)+\nabla V(t,u(t))=0$. One can consider more general case $F(v)=\phi(|v|)$, where $\phi$ is convex and nonnegative. In the above cases $F$ does not depend on $v$ directly but rather on its norm $|v|$ and the growth of $F$ is the same in all directions, i.e. $F$ has isotropic growth. Equation (1) with Lagrangian $L(t,x,v)=\frac{1}{p}|v|^{p}+V(t,x)$ has been studied by many authors under different conditions. The classical reference is [1]. The isotropic Orlicz-Sobolev space setting was considered in [2].

We are interested in anisotropic case. This means that $F$ depends on all components of $v$ not only on $|v|$ and has different growth in different directions. A simple example of such function is $F(v)=\sum_{i=1}^{N}|v_{i}|^{p_{i}}$ or $F(v)=\sum_{i=1}^{N}\phi_{i}(|v_{i}|)$, where $\phi_{i}$ are N-functions. We wish to consider more general situation. We assume that $F\colon[a,b]\times\mathbb{R}^{N}\times\mathbb{R}^{N}\to\mathbb{R}$ satisfies

1. ($F_{1}$)

$F\in C^{1}$, 2. ($F_{2}$)

$|F(t,x,v)|\leq a(|x|)(b(t)+G(v)),$ 3. ($F_{3}$)

$|F_{x}(t,x,v)|\leq a(|x|)(b(t)+G(v)),$ 4. ($F_{4}$)

$G^{\ast}(F_{v}(t,x,v))\leq a(|x|)(c(t)+G^{\ast}(\nabla G(v))).$

where $a\in C(\mathbb{R}_{+},\mathbb{R}_{+})$, $b,c\in\operatorname{\mathbf{L}}^{{1}}(I,\mathbb{R}_{+})$ and $G\colon\mathbb{R}^{N}\to\mathbb{R}$ is a G-function. Conditions $(F_{1})$–$(F_{4})$ are direct generalization of standard growth conditions from [1] (see also [2]). We show (see Theorem 5.7) that under these conditions the functional $\operatorname{\mathcal{I}}\colon{\operatorname{\mathbf{W}}^{1}}\operatorname{\mathbf{L}}^{{G}}\to\mathbb{R}$ given by

[TABLE]

is continuously differentiable.

We restrict our considerations to a special class of G-functions. Here $G\colon\mathbb{R}^{n}\to[0,\infty)$ is convex, $G(-x)=G(x)$, supercoercive, $G(0)=0$ and satisfies $\Delta_{2}$ and $\nabla_{2}$ conditions. We define the anisotropic Orlicz space to be

[TABLE]

The Orlicz space $\operatorname{\mathbf{L}}^{{G}}$ equipped with the Luxemburg norm

[TABLE]

is a reflexive Banach space. An important example of Orlicz space is classical Lebesgue $\operatorname{\mathbf{L}}^{{p}}$ space, defined by $G(x)=\frac{1}{p}|x|^{p}$. In this case, the Luxemburg norm and the standard $\operatorname{\mathbf{L}}^{{p}}$ norm are equivalent. Therefore, Orlicz spaces can be viewed as a straightforward generalization of $\operatorname{\mathbf{L}}^{{p}}$ spaces.

Properties of N-functions and of Orlicz spaces of real-valued functions has been studied in great details in monographs [3, 4, 5] and [6]. The standard references for vector-valued case are [7, 8, 9] and [10, 11] for Banach-space valued functions. In [7, 8] author considers a class of G-functions together with a uniformity conditions which, for example excludes the function $G(x)=\sum|x_{i}|^{p_{i}}$ unless $1<p_{1}=\dots=p_{N}<\infty$. Moreover $G$ is not neccessairly assumed to be an even function. As was pointed out in [11], if $G$ is not even then $\operatorname{\mathbf{L}}^{{G}}$ is no longer a vector space (see also [10, Example 2.1]).

Our strong conditions on $G$ allow us to work in Orlicz spaces without worry about some technical difficulties arising in general case. For example, it is well known that the set $\operatorname{\mathbf{L}}^{{G}}(I,\mathbb{R}^{N})$ is a vector space if and only if $G$ satisfies $\Delta_{2}$ condition. Otherwise $\operatorname{\mathbf{L}}^{{G}}$ is only a convex set. Another difficulty is the convergence notion. In Lebesgue spaces $\|u_{n}-u\|_{\operatorname{\mathbf{L}}^{{p}}}\to 0$ means simply $\int|u_{n}-u|^{p}\to 0$. For arbitrary G-function $G$, convergence in Luxemburg norm is not equivalent to $\int G(u_{n}-u)\,dt\to 0$ unless $G$ satisfies $\Delta_{2}$. The $\Delta_{2}$ condition is also crucial for separability and reflexivity of $\operatorname{\mathbf{L}}^{{G}}$.

The main consequence of anisotropic nature of $G$ is the lack of monotonicity of the norm. It is no longer true that $|u|\leq|v|$ implies $\|u\|_{\operatorname{\mathbf{L}}^{{G}}}\leq\|v\|_{\operatorname{\mathbf{L}}^{{G}}}$. In anisotropic case, standard dominance condition $|u_{n}|\leq f$ does not implies convergence in $\operatorname{\mathbf{L}}^{{G}}$ norm and must be replaced by $G(u_{n})\leq f$ (see Theorem 3.14).

Following [10] we show that for every $G$ we consider there exist $p,q\in(1,\infty)$ such that $\operatorname{\mathbf{L}}^{{q}}\hookrightarrow\operatorname{\mathbf{L}}^{{G}}\hookrightarrow\operatorname{\mathbf{L}}^{{q}}$. If $G(x)=\sum|x_{i}|^{p_{i}}$ then $\operatorname{\mathbf{L}}^{{G}}$ can be identified with the product of $\operatorname{\mathbf{L}}^{{p_{i}}}$ but in many cases an anisotropic Orlicz Space is not equal to the space $\operatorname{\mathbf{L}}^{{p_{1}}}\times\operatorname{\mathbf{L}}^{{p_{2}}}\times...\times\operatorname{\mathbf{L}}^{{p_{N}}}$ (see Example 3.23).

To give a proper variational setting for equation (1) we introduce a notion of an anisotropic Orlicz-Sobolev space ${\operatorname{\mathbf{W}}^{1}}\operatorname{\mathbf{L}}^{{G}}$ of vector-valued functions. It is defined to be

[TABLE]

with the norm

[TABLE]

To the authors best knowledge there is no reference for the case of anisotropic norm and vector-valued functions of one variable. The references for other cases are [2, 9, 12, 13, 14, 15, 16, 17, 18, 19].

In [9] and [18] the space $H^{0}(G,\Omega)$, $\Omega\subset\mathbb{R}^{n}$ is defined as a completion of $C_{0}^{1}(\Omega,\mathbb{R}^{n})$ under norm $\|u\|_{H^{0}(G,\Omega)}=\|Du\|_{G,\Omega}$. It is classical result due to Trudinger $H^{0}(G,\Omega)\hookrightarrow L_{A}(\Omega)$, where $A$ is some N-function (see also Cianchi [14]).

In [17] and [19] the anisotropic Orlicz-Sobolev space $W^{1}L_{G}$ is defined for G-function $G:\mathbb{R}^{n+1}\to[0,\infty]$ as a space of weakly differentiable functions $u:\mathbb{R}^{n}\supset\Omega\to\mathbb{R}$ such that $(u,D_{1}u,D_{2}u,...,D_{n}u)$ belongs to the Orlicz space generated by $G$. A norm for $W^{1}L_{G}$ is given by

[TABLE]

In [12] we can find definition of isotropic Orlicz-Sobolev space of real valued functions

[TABLE]

where $L_{A}$ is Orlicz Space and $A$ is an N-function.

In [2] the isotropic Orlicz-Sobolev space if vector-valued functions is defined to be a space of absolutely continuous functions $u:[0,T]\to\mathbb{R}^{d}$ such that $u$ and $\dot{u}$ belongs to Orlicz space generated by an N-function. Similar treatment can be found in [20].

2. G-functions

Let $\langle\cdot,\cdot\rangle$ denote the standard inner product on $\mathbb{R}^{N}$ and $|\cdot|$ is the induced norm. We assume that $G\colon\mathbb{R}^{N}\to[0,\infty)$ satisfies the following conditions:

1. ($G_{1}$)

$G(0)=0$, 2. ($G_{2}$)

$G$ is convex, 3. ($G_{3}$)

$G$ is even, 4. ($G_{4}$)

$G$ is supercoercive:

[TABLE] 5. ($G_{5}$)

$G$ satisfies the $\Delta_{2}$ condition:

[TABLE] 6. ($G_{6}$)

$G$ satisfies the $\nabla_{2}$ condition:

[TABLE]

A function $G$ is a G-function in the sense of Trudinger [9]. In general, G-function can be unbounded on bounded sets and need not satisfy conditions ($G_{4}$)–($G_{6}$) but only $\lim_{x\to\infty}G(x)=\infty$. A G-function of one variable is called N-function. Some typical examples of $G$ are

1. (1)

$G(x)=\frac{1}{p}|x|^{p}$, $1<p<\infty$ 2. (2)

$G(x)=\sum_{i=1}^{N}G_{p_{i}}(x_{i})$, $1<p_{i}<\infty$ 3. (3)

$G(x)=(x_{1}-x_{2})^{2}+x_{2}^{4}$

A function $G$ can be equal to zero in some neighborhood of [math]. So that a function

[TABLE]

is also admissible. Conditions $\Delta_{2}$ and $\nabla_{2}$ implies that $G$ is of polynomial growth (see Lemma 2.4 below and [3]). A function $f:\mathbb{R}^{2}\to\mathbb{R}$ $f(x)=e^{|x|}-|x|-1$ does not satisfy $\Delta_{2}$.

Since $G$ is convex and finite on $\mathbb{R}^{n}$, $G$ is locally Lipschitz and therefore continuous. Note that for every $x\in\mathbb{R}^{N}$

[TABLE]

We obtain immediately that $G$ is non-decreasing along any half-line through the origin i.e. for every $x\in\mathbb{R}^{N}$

[TABLE]

Our assumptions on $G$ imply that for every $x_{0}\in\mathbb{R}^{N}$ there exists $a\in\mathbb{R}^{N}$ and $b\in\mathbb{R}$ such that for all $x\in\mathbb{R}^{N}$

[TABLE]

From this, we can easily obtain the Jensen integral inequality. Let $I\subset\mathbb{R}$ be a finite interval and let $u\in\operatorname{\mathbf{L}}^{{1}}(I,\mathbb{R}^{N})$. Then

[TABLE]

We will often make use of the following simple observation.

Proposition 2.1**.**

For all $\alpha\in\mathbb{R}$ there exists $K_{1}(\alpha)>0$ such that

[TABLE]

for all $|x|\geq M_{1}$.

In fact, the above proposition provides a characterization of $\Delta_{2}$ (see [7, 11]). It follows that for every $\alpha\in\mathbb{R}$ there exists $C_{\alpha}>0$ such that for $x\in\mathbb{R}^{N}$

[TABLE]

We recall a notion of Fenchel conjugate. Define $G^{\ast}:\mathbb{R}^{N}\to[0,\infty)$ by

[TABLE]

A function $G^{\ast}$ is called Fenchel conjugate of $G$. As an immediate consequence of definition we have the so called Fenschel inequality:

[TABLE]

Consider arbitrary $f\colon\mathbb{R}^{N}\to[0,\infty)$. It is obvious that the conjugate function $f^{\ast}$ is always convex. But in general $f^{\ast}$ need not be continuous, finite or coercive, even if $f$ is. From the other hand, it is well known that if $f$ is convex and l.s.c. then $f^{\ast}\not\equiv\infty$ and $(f^{\ast})^{\ast}=f$.

Example 2.2**.**

1. (1)

If

[TABLE]

then $g^{\ast}(x)=|x|.$ Note that $g$ and $g^{\ast}$ are G-functions but do not satisfy our assumptions. 2. (2)

If $G(x)=\frac{1}{p}|x|^{p}$, then $G^{\ast}(x)=\frac{1}{q}|x|^{q}$, $\frac{1}{p}+\frac{1}{q}=1$. 3. (3)

If $G(x)=\sum_{i=1}^{N}G_{i}(x_{i})$ then $G^{\ast}(x)=\sum_{i=1}^{N}G_{i}^{\ast}(x_{i})$. 4. (4)

If $G(x,y)=(x-y)^{2}+y^{4}$, then

[TABLE]

More information on general theory of conjugate functions can be found in standard books on convex analysis, see for instance [21, 22].

If a function $G\colon\mathbb{R}^{n}\to[0,\infty)$ satisfies conditions ($G_{1}$)–($G_{6}$) then the same is true for its conjugate $G^{\ast}$. This is main reason we want to restrict class of considered functions.

Theorem 2.3**.**

If $G$ satisfies conditions ($G_{1}$)–($G_{6}$) then $G^{\ast}$ also satisfies ($G_{1}$)–($G_{6}$) and $(G^{\ast})^{\ast}=G$.

Proof.

It is evident that $G^{\ast}$ satisfies ($G_{1}$), ($G_{2}$) and ($G_{3}$). It is well known that under our conditions, $G^{\ast}$ is finite (proposition 1.3.8, [21]), $G^{\ast}$ is supercoercive (proposition 1.3.9, [21]) and $G^{\ast}$ satisfies ($G_{5}$) and ($G_{6}$) (remark 2.3, [10]). Corrollary [21, cor. 1.3.6] gives $(G^{\ast})^{\ast}=G$. ∎

In order to compare growth rate of G-functions we define two relations. Let $G_{1}$ and $G_{2}$ be G-functions. Define

[TABLE]

and

[TABLE]

For conjugate functions we have (see [3, thm. 3.1])

[TABLE]

Obviously $G_{1}\prec\prec G_{2}$ implies $G_{1}\prec G_{2}$. Assumption ($G_{4}$) implies $|x|\prec\prec G$. It is true that $|x|\prec G$ holds under weaker assumption: $G(x)\to\infty$. Note that, if $p>1$ then $|x|\prec\prec|x|^{p}$. Hence, if $|x|^{p}\prec G$ then $|x|\prec\prec G$. Since $G$ satisfies ($G_{5}$) and ($G_{6}$) we have the following bounds for the growth of $G$.

Lemma 2.4** (cf. [10, Lemma 2.4]).**

There exists $p,q\in(1,\infty)$ such that

[TABLE]

Proof.

Set $C=\overline{G}(M_{1})$. By induction, if $|x|\leq 2^{n}M_{1}$ then $G(x)\leq K_{1}^{n}C$. For $|x|\geq M_{1}$ choose $n$ such that $2^{n-1}M_{1}\leq|x|\leq 2^{n}M_{1}$. Then $n-1\leq\log_{2}(|x|/M_{1})$ and $G(x)\leq CK_{1}^{1+\log_{2}(|x|/M_{1})}$. Therefore, for $|x|\geq M_{1}$,

[TABLE]

This proves that $G\prec|x|^{q}$. Choose $r>0$ such that if $x\in G^{-1}(\overline{G}(M_{1}))$ then $|x|\leq r$. Set $M=rM_{1}$. Again, by induction, for $|x|\geq K_{2}^{k}M$ we have $(2K_{2})^{k}C\leq G(x)$. This implies

[TABLE]

whenever $|x|\geq MK_{2}$. Hence $|x|^{p}\prec G$. ∎

Immediately from the above we get $|x|^{\frac{q}{q-1}}\prec G^{\ast}\prec|x|^{\frac{p}{p-1}}$.

3. Orlicz spaces

Let $I\subset\mathbb{R}$ be a finite interval. The Orlicz space $\operatorname{\mathbf{L}}^{{G}}=\operatorname{\mathbf{L}}^{{G}}(I,\mathbb{R}^{n})$ is defined to be

[TABLE]

As usual, we identify functions equal a.e. For an arbitrary G-function $f\colon\mathbb{R}^{n}\to[0,\infty)$ which does not satisfies $\Delta_{2}$ the set $\operatorname{\mathbf{L}}^{{f}}$ is not a linear space but only a convex set. In fact, it is well known that the set $\operatorname{\mathbf{L}}^{{f}}$ is linear space if and only if a G-function $f$ satisfies $\Delta_{2}$ condition.

For $u\in\operatorname{\mathbf{L}}^{{G}}$ define:

[TABLE]

The function $\|\cdot\|_{\operatorname{\mathbf{L}}^{{G}}}$ is called the Luxemburg norm. It is easy to see that

[TABLE]

since $G$ satisfies $\Delta_{2}$. Moreover

[TABLE]

Remark 3.1**.**

All properties of $\operatorname{\mathbf{L}}^{{G}}$ remains true for $\operatorname{\mathbf{L}}^{{G^{\ast}}},$ since $G$ and $G^{\ast}$ belongs to the same class of functions.

Theorem 3.2**.**

If $G\colon\mathbb{R}^{n}\to[0,\infty)$ satisfies ($G_{1}$)–($G_{6}$), then $(\operatorname{\mathbf{L}}^{{G}}(I,\mathbb{R}^{n}),\|\cdot\|_{\operatorname{\mathbf{L}}^{{G}}})$ is a normed linear space.

Proof.

We first prove that $\operatorname{\mathbf{L}}^{{G}}$ is a linear space. Since $G$ is continuous and satisfies $\Delta_{2}$, we get

[TABLE]

where $I_{1}={\{t\in I\colon|u(t)|\leq M_{1}\}}$. Hence, if $u\in\operatorname{\mathbf{L}}^{{G}}$ then $\alpha u\in\operatorname{\mathbf{L}}^{{G}}$ for all $\alpha\in\mathbb{R}$. For every $u,v\in\operatorname{\mathbf{L}}^{{G}}$ and $\alpha,\beta\in\mathbb{R}$, by ($G_{2}$) and Proposition 2.1, we have

[TABLE]

Hence $\alpha u+\beta v\in\operatorname{\mathbf{L}}^{{G}}$.

Now we show that $\|\cdot\|_{\operatorname{\mathbf{L}}^{{G}}}$ is a norm on $\operatorname{\mathbf{L}}^{{G}}$. It is evident that if $u=0$ then $\|u\|_{\operatorname{\mathbf{L}}^{{G}}}=0$. Suppose $u\neq 0$. There exists $I_{1}\subset I$ with positive measure and $\varepsilon>0$ such that for all $t\in I_{1}$, $|u(t)|\geq\varepsilon$. For every $t\in I_{1}$ there exists $\alpha_{t}\geq 1$ and $y_{t}\in\mathbb{R}^{n}$, $|y_{t}|=\varepsilon$ such that $u(t)=\alpha_{t}y_{t}$. For all $k>0$ we have by (2) that $G(\alpha_{t}y_{t}/k)\geq G(y_{t}/k)$. Hence

[TABLE]

where $\underline{G}(\varepsilon/k)=\inf\{G(y)\colon|y|=\frac{\varepsilon}{k}\}$. Since $\underline{G}(\varepsilon/k)\nearrow\infty$ as $k\searrow 0$, there exists $k_{0}>0$ such that for all $k\leq k_{0}$

[TABLE]

and

[TABLE]

Finally, $\|u\|_{\operatorname{\mathbf{L}}^{{G}}}=0\iff u=0$. Let $u\in L^{G}$ and $\alpha\in\mathbb{R}$. For $\alpha\in\mathbb{R}$:

[TABLE]

If $\|u\|_{\operatorname{\mathbf{L}}^{{G}}}=0$ or $\|v\|_{\operatorname{\mathbf{L}}^{{G}}}=0$, then it is obvious that $\|u+v\|_{\operatorname{\mathbf{L}}^{{G}}}\leq\|u\|_{\operatorname{\mathbf{L}}^{{G}}}+\|v\|_{\operatorname{\mathbf{L}}^{{G}}}$. Set $\alpha=\|u\|_{\operatorname{\mathbf{L}}^{{G}}}>0$, $\beta=\|v\|_{\operatorname{\mathbf{L}}^{{G}}}>0$. Then $\int_{I}G\left(\frac{u}{\alpha}\right)=1$ and $\int_{I}G\left(\frac{v}{\beta}\right)=1$. Thus

[TABLE]

As a consequence

[TABLE]

∎

An important example of Orlicz space is a classical Lebesgue space $(\operatorname{\mathbf{L}}^{{p}},\|\cdot\|_{\operatorname{\mathbf{L}}^{{p}}})$, $p\in(1,\infty)$ defined by $G(x)=\frac{1}{p}|x|^{p}$. It is easy to check that in this case $\operatorname{\mathbf{L}}^{{G}}=\operatorname{\mathbf{L}}^{{p}}$ and the Luxemburg norm and standard $\operatorname{\mathbf{L}}^{{p}}$ norm are equivalent. Two important examples of Lebesgue spaces are not covered in our setting, namely $\operatorname{\mathbf{L}}^{{1}}$ and $\operatorname{\mathbf{L}}^{{\infty}}$. The space $\operatorname{\mathbf{L}}^{{1}}$ is generated by $f(x)=|x|$ and the space $\operatorname{\mathbf{L}}^{{\infty}}$ generated by $f^{\ast}$. We exclude these two spaces because we want to have only reflexive spaces in the class of Orlicz spaces we consider.

It was pointed out by Schappacher [11, example 3.1] that if $f$ is not bounded on bounded sets (i.e. we allow $f(x)=+\infty$ for some $x\in\mathbb{R}^{n}$) then $\operatorname{\mathbf{L}}^{{f}}$ need not be a linear space, even if $f$ satisfies $\Delta_{2}$ condition. To see this, consider

[TABLE]

See [3, 11] for more details.

Theorem 3.3** (Hölder inequality).**

For every $u\in\operatorname{\mathbf{L}}^{{G}}$ and $v\in\operatorname{\mathbf{L}}^{{G^{\ast}}}$

[TABLE]

Proof.

Using Fenchel inequality we obtain

[TABLE]

Hence

[TABLE]

∎

We finish this section by completeness of Orlicz space.

Theorem 3.4** (cf. [3], [11, theorem 6.1]).**

The space $(\operatorname{\mathbf{L}}^{{G}},\|\cdot\|_{\operatorname{\mathbf{L}}^{{G}}})$ is complete.

Proof.

Let $\{u_{n}\}$ be a Cauchy sequence in $\operatorname{\mathbf{L}}^{{G}}$. Fix $\delta,\epsilon>0$ and choose $\alpha>0$ such that $G(\alpha x)>2/\delta$ if $|x|\geq\varepsilon$. Let $n_{0}$ be large enough so that $\|u_{n}-u_{m}\|_{\operatorname{\mathbf{L}}^{{G}}}\leq\alpha^{-1}$, i.e

[TABLE]

Put $E=\{t\colon G(\alpha(u_{n}(t)-u_{m}(t)))>\delta/2\}.$ Then

[TABLE]

that is $\mu(E)<\frac{\delta}{2}$. It follows that

[TABLE]

Thus $\{u_{n}\}$ is a Cauchy sequence in measure. This follows that there is a subsequence $\{u_{n_{k}}\}$ convergent a.e. to some measurable function $u$.

Fix $\varepsilon>0$ and choose $K$ such that for $k,l>K$, $\|u_{n_{k}}-u_{n_{l}}\|_{\operatorname{\mathbf{L}}^{{G}}}\leq\epsilon$. Then

[TABLE]

Letting $n_{l}\to\infty$ we obtain by Fatou Lemma,

[TABLE]

Hence $u_{n_{k}}-u\in\operatorname{\mathbf{L}}^{{G}}$ and consequently $u\in\operatorname{\mathbf{L}}^{{G}}$. Since $\varepsilon>0$ is arbitrary, $\|u_{n_{k}}~{}-~{}u\|_{\operatorname{\mathbf{L}}^{{G}}}\to 0$ and $\|u_{n}-u\|_{\operatorname{\mathbf{L}}^{{G}}}\to 0$. ∎

3.1. Convergence

Now we investigate relations between Luxemburg norm and the integral

[TABLE]

A functional $R_{G}$ is called modular. Theory of modulars is well known and is developed in more general setting than ours. More information can be found in [23, 5].

For Lebesgue spaces a notions of modular and norm are indistinguishable because modular $\int_{I}|u|^{p}\,dt$ is equal to $\|u\|_{\operatorname{\mathbf{L}}^{{p}}}^{p}$. But in Orlicz spaces relation between $R_{G}$ and $\|\cdot\|_{\operatorname{\mathbf{L}}^{{G}}}$ is more complex.

There is remarkable difference between isotropic and anisotropic spaces. It is clear that if $u,v\in\operatorname{\mathbf{L}}^{{p}}$ (or more generally in isotropic Orlicz space) then $|u(t)|\leq|v(t)|$ a.e. implies $\|u\|_{\operatorname{\mathbf{L}}^{{p}}}\leq\|v\|_{\operatorname{\mathbf{L}}^{{p}}}$. In anisotropic case it is no longer true, even if $G(u(t))<G(v(t))$. Next two examples illustrates this point.

Example 3.5**.**

Let $G(x,y)=(x-y)^{2}+y^{4}$, $I=[0,1]$, $u(t)=(2,0)$ and $v(t)=(2,3/2)$. Then $|u(t)|<|v(t)|$, $G(u(t))<G(v(t))$ and $R_{G}(u)\leq R_{G}(v)$, but $2=\|u\|_{\operatorname{\mathbf{L}}^{{G}}}>\|v\|_{\operatorname{\mathbf{L}}^{{G}}}\simeq 1.6$.

Example 3.6**.**

Let $G(x,y)=x^{2}+y^{4}$, $u(t)=(1,0)$ and $v(t)=\tfrac{11}{10}(\cos t,\sqrt{\sin t})$. In $\operatorname{\mathbf{L}}^{{G}}([0,\pi],\mathbb{R}^{2})$ we have

[TABLE]

but $|u(t)|<|v(t)|$, $G(u(t))<G(v(t))$ for all $t\in[0,\pi]$ and $R_{G}(u)<R_{G}(v)$.

Definition 3.7**.**

We say that a subset $K\subset\operatorname{\mathbf{L}}^{{G}}$ is modular bounded if there exists $C>0$ such that

[TABLE]

Modular boundedness is sometimes called mean boundedness. It is evident that $R_{G}(u)\leq\|u\|_{\operatorname{\mathbf{L}}^{{G}}}$ if $\|u\|_{\operatorname{\mathbf{L}}^{{G}}}\leq 1$ and $R_{G}(u)>\|u\|_{\operatorname{\mathbf{L}}^{{G}}}$ if $\|u\|_{\operatorname{\mathbf{L}}^{{G}}}>1$.

Lemma 3.8**.**

Let $u\in\operatorname{\mathbf{L}}^{{G}}$.

1. (1)

If $R_{G}(u)\leq C$ then $\|u\|_{\operatorname{\mathbf{L}}^{{G}}}\leq\max\{C,1\}$. 2. (2)

If $\|u\|_{\operatorname{\mathbf{L}}^{{G}}}\leq C$ then $R_{G}(u)\leq\mu(I)\widetilde{C}+K_{1}(C)$ for some $\widetilde{C}>0$.

Moreover, a set $K\subset\operatorname{\mathbf{L}}^{{G}}$ is modular bounded if and only if is norm bounded.

Proof.

Assume that $R_{G}(u)\leq C$. If $C\leq 1$ then $\|u\|_{\operatorname{\mathbf{L}}^{{G}}}\leq 1$. If $C>1$ then

[TABLE]

This implies $\|u\|_{\operatorname{\mathbf{L}}^{{G}}}\leq\max\{C,1\}$. For the second statement, assume $\|u\|_{\operatorname{\mathbf{L}}^{{G}}}\leq~{}C$. Then

[TABLE]

where $I_{1}=\{t\in I\colon|u(t)|\leq M_{1}C\}$ and $\widetilde{C}>0$. To finish the proof observe that

[TABLE]

∎

Definition 3.9**.**

We say that a sequence of functions $u_{k}\in\operatorname{\mathbf{L}}^{{G}}$ is modular convergent to $u\in\operatorname{\mathbf{L}}^{{G}}$ if $R_{G}(u_{k}-u)\to 0$ as $k\to\infty$.

Modular convergence is sometimes called mean convergence. Norm convergence always implies modular convergence. Let $\|u_{k}\|_{\operatorname{\mathbf{L}}^{{G}}}\to 0$ as $k\to\infty$. We can assume that $\forall_{k}\ \|u_{k}\|_{\operatorname{\mathbf{L}}^{{G}}}\leq 1$, then

[TABLE]

Hence $0\leq R_{G}(u_{k})\leq\|u_{k}\|_{\operatorname{\mathbf{L}}^{{G}}}$. In general, converse is not true unless $G$ satisfies $\Delta_{2}$ condition. (see [3, 11]).

Theorem 3.10**.**

Norm convergence is equivalent to modular convergence.

Proof.

We need only to prove that modular convergence implies norm convergence. Fix $\varepsilon>0$ and assume that $\{u_{k}\}$ is modular convergent to [math]. Define

[TABLE]

Since $G$ satisfies $\Delta_{2}$, for all $k>0$ we have

[TABLE]

For sufficiently large $k$ we have

[TABLE]

and

[TABLE]

Finally, Lemma 3.8 shows that $\|u_{k}\|_{\operatorname{\mathbf{L}}^{{G}}}\leq C\varepsilon$ and hence $\|u_{k}\|_{\operatorname{\mathbf{L}}^{{G}}}\to 0$. ∎

It is standard result due to Riesz that for $f_{n}$, $f\in\operatorname{\mathbf{L}}^{{p}}$

[TABLE]

Following lemmas establish Orlicz space version of this fact.

Lemma 3.11**.**

For every $k>1$ and $0<\varepsilon<\frac{1}{k}$ and $x,y\in\mathbb{R}^{n}$

[TABLE]

where $C_{\varepsilon}=\frac{1}{\varepsilon(k-1)}$

Proof.

The proof is due to Brezis and Lieb [24] (see also [25]). We repeat the proof. Let $\alpha=1-k\varepsilon$, $\beta=\varepsilon$, $\gamma=\varepsilon(k-1)$. Then $\alpha+\beta+\gamma=1$ and $x+y=\alpha x+\beta(kx)+\gamma(C_{\varepsilon}y)$. By convexity

[TABLE]

This implies that

[TABLE]

For the reverse inequality let

[TABLE]

Then $x=\alpha(x+y)+\beta(kx)+\gamma(-C_{\varepsilon}y)$ and

[TABLE]

∎

Lemma 3.12**.**

If $u_{n}\to u$ in $\operatorname{\mathbf{L}}^{{G}}$ then $R_{G}(u_{n})\to R_{G}(u)$.

Proof.

In Lemma 3.11 set $x+y=u_{n}$, $x=u$, $k=2$. Then $\varepsilon<1/2$, $C_{\varepsilon}=\frac{1}{\varepsilon}$ and

[TABLE]

Since $u_{n}\to u$ in $\operatorname{\mathbf{L}}^{{G}}$, there exists $n_{0}$ such that for $n>n_{0}$ we have $\|u_{n}-u\|_{\operatorname{\mathbf{L}}^{{G}}}<\varepsilon^{2}\leq\varepsilon<1$. Thus

[TABLE]

From this and inequality above we obtain

[TABLE]

Letting $\varepsilon\to 0$ we have $R_{G}(u_{n})\to R_{G}(u)$. ∎

Norm convergence $u_{n}\to u$ in $\operatorname{\mathbf{L}}^{{p}}$ implies that there exists a subsequence such that $u_{n_{k}}\to u$ a.e. and $|u_{n_{k}}|\leq|h|\in\operatorname{\mathbf{L}}^{{p}}$. According to the above lemma, if $u_{n}\to u$ in $\operatorname{\mathbf{L}}^{{G}}$ then:

1. (1)

Since $\operatorname{\mathbf{L}}^{{G}}\hookrightarrow\operatorname{\mathbf{L}}^{{1}}$ (see Lemma 3.20 below), we can extract a subsequence $u_{n_{k}}$ such that

[TABLE] 2. (2)

Since $R_{G}(u_{n}-u)\to 0$, $G(u_{n}-u)\to 0$ in $\operatorname{\mathbf{L}}^{{1}}$. Thus we can extract a subsequence $\{u_{n_{k}}\}$ such that

[TABLE] 3. (3)

Since $R_{G}(u_{n})\to R_{G}(u)$, $G(u_{n})\to G(u)$ in $\operatorname{\mathbf{L}}^{{1}}$. Hence there exists a subsequence $\{u_{n_{k}}\}$ such that

[TABLE]

Lemma 3.13**.**

Let $\{u_{n}\}\subset\operatorname{\mathbf{L}}^{{G}}$ and $u\in\operatorname{\mathbf{L}}^{{G}}$. Suppose that

1. (1)

$u_{n}\to u$ a.e. 2. (2)

$R_{G}(u_{n})\to R_{G}(u)$.

Then $u_{n}\to u$ in $\operatorname{\mathbf{L}}^{{G}}$.

Proof.

This lemma was proved in [4, p. 83] for N-functions. Since $G$ is convex, we get $\frac{1}{2}(G(u_{n}(t))+G(u(t)))-G(\frac{u_{n}(t)-u(t)}{2})\geq 0$. Continuity of $G$ and $u_{n}\to u$ a.e. implies

[TABLE]

So that by the Fatou Lemma, we have

[TABLE]

This implies that

[TABLE]

and $\|u_{k}-u\|_{\operatorname{\mathbf{L}}^{{G}}}\to 0$ by Theorem 3.10. ∎

As a consequence we obtain dominated convergence theorem for anisotropic Orlicz spaces:

Theorem 3.14**.**

Suppose that $\{u_{n}\}\subset\operatorname{\mathbf{L}}^{{G}}$ and

1. (1)

$u_{n}\to u$ a.e. 2. (2)

there exists $h\in\operatorname{\mathbf{L}}^{{1}}$ such that $G(u_{n})\leq h$ a.e.

Then $u\in\operatorname{\mathbf{L}}^{{G}}$ and $u_{n}\to u$ in $\operatorname{\mathbf{L}}^{{G}}$.

Proof.

Since $G$ is continuous and $u_{n}\to u$ a.e., $G(u_{n})\to G(u)$ a.e. It follows that $G(u)\leq h$ a.e. Thus $G(u)\in\operatorname{\mathbf{L}}^{{1}}$ and hence $u\in\operatorname{\mathbf{L}}^{{G}}$. Since $0\geq h\pm G(u_{n})$ and $h\pm G(u_{n})\to h\pm G(u)$ a.e., application of the Fatou Theorem yields

[TABLE]

Therefore,

[TABLE]

and hence

[TABLE]

and $R_{G}(u_{n})\to R_{G}(u)$. By the Lemma 3.13, $u_{n}\to u$ in $\operatorname{\mathbf{L}}^{{G}}$. ∎

In the above theorem, assumption $G(u_{n})\leq h$ can be replaced by $G(u_{n})\leq G(h)$, $h\in\operatorname{\mathbf{L}}^{{G}}$. Consider a sequence $\{u_{n}\}\subset\operatorname{\mathbf{L}}^{{G}}$ convergent pointwise to measurable function $u$. Under standard dominance condition (i.e. $|u_{n}|\leq|g|$, $g\in\operatorname{\mathbf{L}}^{{G}}$) it is not true in general that $u_{n}\to u\in\operatorname{\mathbf{L}}^{{G}}$.

Example 3.15**.**

Let $G(x,y)=x^{2}+y^{4}$, $I=(0,1)$, $u(t)=(0,t^{-1/4})\text{ and }h(t)=(t^{-3/8},0)$. Define

[TABLE]

Then $u_{n}\to u$ a.e., $u_{n},h\in\operatorname{\mathbf{L}}^{{G}}$ and $|u_{n}|\leq|h|$ for every $t$. But $G(u(t))=t^{-1}\notin\operatorname{\mathbf{L}}^{{1}}(I,\mathbb{R})$. Hence $u\notin\operatorname{\mathbf{L}}^{{G}}$.

Remark 3.16**.**

Modular $R_{G}$ is called monotone modular if $|x|\leq|y|$ implies $R_{G}(x)\leq R_{G}(y)$. If $R_{G}$ is monotone modular then $u_{k}\to u$ a.e and $|u_{k}|\leq|g|$, $g\in\operatorname{\mathbf{L}}^{{G}}$ implies $u\in\operatorname{\mathbf{L}}^{{G}}$ and $\|u_{k}-u\|_{\operatorname{\mathbf{L}}^{{G}}}\to 0$. We refer the reader to [25] for more details.

3.2. Separability

For every $u\in\operatorname{\mathbf{L}}^{{G}}$ there exists a sequence of bounded functions $\{u_{n}\}\subset\operatorname{\mathbf{L}}^{{G}}$ such that $u_{n}\to u$ in $\operatorname{\mathbf{L}}^{{G}}$. For example, one can define

[TABLE]

In this case $u_{n}\to u$ a.e and $G(u_{n}(t)-u(t))\leq G(u(t))$. Therefore, by Theorem 3.14 we get $u_{n}\to u$ in $\operatorname{\mathbf{L}}^{{G}}$.

Theorem 3.17** (cf. [3, p. 81]).**

The space $\operatorname{\mathbf{L}}^{{G}}$ is separable.

Proof.

Fix $\varepsilon>0$. Suppose that $u\in\operatorname{\mathbf{L}}^{{G}}$ is bounded and $|u(t)|\leq a$. Set $C=\sup\{G(x/\varepsilon)\colon|x|\leq 2a\}$.

By the Luzin theorem we can find a compact subset $I_{1}\subset I$ and a continuous function $u_{1}\colon I\to\mathbb{R}^{N}$ such that $\mu(I\setminus I_{1})\leq 1/C$, $u(t)=u_{1}(t)$ for all $t\in I_{1}$ and $|u_{1}(t)|\leq a$. Now we get

[TABLE]

Hence $\|u-u_{1}\|_{\operatorname{\mathbf{L}}^{{G}}}\leq\varepsilon$. For arbitrary $v\in\operatorname{\mathbf{L}}^{{G}}$ we can find a bounded $u_{1}\in\operatorname{\mathbf{L}}^{{G}}$ such that $\|v-u\|_{\operatorname{\mathbf{L}}^{{G}}}\leq\varepsilon/2$. Thus

[TABLE]

For every continuous function there exists uniformly convergent sequence of polynomials with rational coefficients. It is easy to check that uniform convergence implies norm convergence in $\operatorname{\mathbf{L}}^{{G}}$. This completes the proof. ∎

Remark 3.18**.**

It is well known that if G-function does not satisfies $\Delta_{2}$ condition then $\operatorname{\mathbf{L}}^{{G}}$ is not separable. One can define a subspace $E^{G}$ as the closure of bounded functions under Luxemburg norm. In this case, the space $E^{G}$ is a proper subset of $\operatorname{\mathbf{L}}^{{G}}$ and is always separable (see [3, 11]).

3.3. Embeddings

We will use the symbols $\hookrightarrow$ nad $\hookrightarrow\hookrightarrow$ for, respectively, continuous and compact embeddings. Recall that

[TABLE]

and

[TABLE]

Next two theorems provide a basic embeddings for Orlicz spaces.

Proposition 3.19**.**

Assume that $F\prec G$. Then $L^{G}\hookrightarrow L^{F}$ and

[TABLE]

for some $C>0$.

Proof.

It is evident that $\operatorname{\mathbf{L}}^{{G}}\subset\operatorname{\mathbf{L}}^{{F}}$. Let $u\in\operatorname{\mathbf{L}}^{{G}}$ and set

[TABLE]

For every $t\in I_{1}$, we have

[TABLE]

and

[TABLE]

where $\widetilde{C}=\sup\{G(x)\colon|x|\leq M\}$. Since $1\leq\widetilde{C}\mu(I)+1$, we have

[TABLE]

Finally,

[TABLE]

∎

It is easy to see that there exist constants $C_{1},C_{2}>0$ such that $\|u\|_{\operatorname{\mathbf{L}}^{{1}}}\leq C_{1}\|u\|_{\operatorname{\mathbf{L}}^{{G}}}$ and $\|u\|_{\operatorname{\mathbf{L}}^{{G}}}\leq C_{2}\|u\|_{\operatorname{\mathbf{L}}^{{\infty}}}$.

Directly from Lemma 2.4 we obtain that Orlicz spaces can be viewed as a spaces between two Lebesgue spaces determined by constants in $\Delta_{2}$ and $\nabla_{2}$ conditions.

Proposition 3.20**.**

For every $G$ there exists $p,q\in(1,\infty)$ such that

[TABLE]

In particular $\operatorname{\mathbf{L}}^{{\infty}}\hookrightarrow\operatorname{\mathbf{L}}^{{G}}\hookrightarrow\hookrightarrow\operatorname{\mathbf{L}}^{{1}}$.

Theorem 3.21** (cf. [6, th. 8.25]).**

If $F\prec\prec G$ then $\operatorname{\mathbf{L}}^{{G}}\hookrightarrow\hookrightarrow\operatorname{\mathbf{L}}^{{F}}$.

Proof.

Let $\{u_{n}\}$ be a bounded sequence in $\operatorname{\mathbf{L}}^{{G}}$. Since $\operatorname{\mathbf{L}}^{{G}}\hookrightarrow\operatorname{\mathbf{L}}^{{F}}\hookrightarrow\hookrightarrow\operatorname{\mathbf{L}}^{{1}}$, $\{u_{n}\}$ is bounded in $\operatorname{\mathbf{L}}^{{F}}$ and there exists a subsequence, denoted again by $\{u_{n}\}$, convergent in $\operatorname{\mathbf{L}}^{{1}}$. Hence $\{u_{n}\}$ converges in measure and thus is Cauchy in measure.

Fix $\varepsilon>0$ and let $v_{n,k}(t)=(u_{n}(t)-u_{k}(t))/\varepsilon$. Since $\{u_{n}\}$ is bounded in $\operatorname{\mathbf{L}}^{{F}}$, there exist $C>0$ such that $\|u_{n}\|_{\operatorname{\mathbf{L}}^{{F}}}\leq C$. There exist $M>0$ such that if $|x|\geq M$, then

[TABLE]

Set $\overline{F}(M)=\sup\{F(x)\colon|x|\leq M\}$,

[TABLE]

Since $\{u_{n}\}$ is Cauchy in measure, there exists $N$ such that if $n$, $k\geq N$, then $\mu(I^{\prime\prime}_{n,k})\leq\mu(I_{n,k})\leq\frac{1}{2\overline{F}(M)}$. Observe that

1. (1)

if $t\in I\setminus I_{n,k}$ then $F(v_{n,k}(t))\leq 1/2\mu(I)$, 2. (2)

if $t\in I^{\prime}_{n,k}$, then $F(v_{n,k}(t))\leq\tfrac{1}{4}G(v_{n,k}/C)$, 3. (3)

if $t\in I^{\prime\prime}_{n,k}$, then $F(v_{n,k}(t))\leq\overline{F}(M)$.

It follows that for $n$, $k\geq N$, we have

[TABLE]

Hence $\|u_{n}-u_{k}\|_{\operatorname{\mathbf{L}}^{{F}}}\leq\varepsilon$ and so $\{u_{n}\}$ converges in $\operatorname{\mathbf{L}}^{{F}}$. ∎

In some cases, $\operatorname{\mathbf{L}}^{{G}}$ is simply a product of $\operatorname{\mathbf{L}}^{{p_{i}}}(I,\mathbb{R})$, but there exists Orlicz spaces which are not in the form $\operatorname{\mathbf{L}}^{{p}}(I,\mathbb{R})\times\operatorname{\mathbf{L}}^{{q}}(I,\mathbb{R})$ (cf. [9, pp. 18-20]).

Example 3.22**.**

Consider the Orlicz space $\operatorname{\mathbf{L}}^{{G}}=\operatorname{\mathbf{L}}^{{G}}(I,\mathbb{R}^{2})$ generated, by $G(x)=|x_{1}|^{p_{1}}+|x_{2}|^{p_{2}}$, $p_{1},p_{2}>0$. If $u=(u_{1},u_{2})\in\operatorname{\mathbf{L}}^{{p_{1}}}(I,\mathbb{R})\times\operatorname{\mathbf{L}}^{{p_{2}}}(I,\mathbb{R})$, then

[TABLE]

Conversely, if $u=(u_{1},u_{2})\in\operatorname{\mathbf{L}}^{{G}}$ then

[TABLE]

Hence $u\in\operatorname{\mathbf{L}}^{{p_{1}}}(I,\mathbb{R})\times\operatorname{\mathbf{L}}^{{p_{2}}}(I,\mathbb{R})$.

Example 3.23**.**

Consider the Orlicz space $\operatorname{\mathbf{L}}^{{G}}=\operatorname{\mathbf{L}}^{{G}}(I,\mathbb{R}^{2})$ generated, by $G(x)=(x_{1}-x_{2})^{4}+x_{2}^{2}$. From Lemmas 2.4 and 3.20 we obtain that $\operatorname{\mathbf{L}}^{{4}}(I,\mathbb{R}^{2})\hookrightarrow\operatorname{\mathbf{L}}^{{G}}\hookrightarrow\operatorname{\mathbf{L}}^{{2}}(I,\mathbb{R}^{2})$. Let $u_{1}$ be a function in $\operatorname{\mathbf{L}}^{{2}}(I,\mathbb{R})$ such that $u_{1}\notin\operatorname{\mathbf{L}}^{{p}}(I,\mathbb{R})$, for $p>2$. Set $u=(u_{1},u_{1})$, then

[TABLE]

but

[TABLE]

Therefore for every $p>2$ there exists $u\in\operatorname{\mathbf{L}}^{{G}}$ such that $u\notin\operatorname{\mathbf{L}}^{{p}}(I,\mathbb{R}^{2})$. Moreover, $u\notin\operatorname{\mathbf{L}}^{{p}}(I,\mathbb{R})\times\operatorname{\mathbf{L}}^{{2}}(I,\mathbb{R})$ for any $p>2$. From the other hand if $u=(u_{1},u_{2})\in\operatorname{\mathbf{L}}^{{4}}(I,\mathbb{R})\times\operatorname{\mathbf{L}}^{{4}}(I,\mathbb{R})$ then $u\in\operatorname{\mathbf{L}}^{{G}}$. Therefore

[TABLE]

but $\operatorname{\mathbf{L}}^{{G}}$ cannot be identified with any

[TABLE]

3.4. Duality

Since $\operatorname{\mathbf{L}}^{{G}}\hookrightarrow\operatorname{\mathbf{L}}^{{p}}\hookrightarrow\hookrightarrow\operatorname{\mathbf{L}}^{{p_{0}}}\hookrightarrow\operatorname{\mathbf{L}}^{{1}}$ ($p$ given by $\nabla_{2}$) and $1<p_{0}<p$, it follows that $\operatorname{\mathbf{L}}^{{G}}$ is closed subspace of reflexive space. Therefore $\operatorname{\mathbf{L}}^{{G}}$ is reflexive itself.

Theorem 3.24**.**

$\operatorname{\mathbf{L}}^{{G}}$ is a reflexive Banach space.

The rest of this section is devoted to proving that the general formula for bounded linear operator $F\colon\operatorname{\mathbf{L}}^{{G}}\to\mathbb{R}$ is

[TABLE]

where $v\in\operatorname{\mathbf{L}}^{{G^{\ast}}}$. We show that the dual space $(\operatorname{\mathbf{L}}^{{G}})^{\ast}$ can be identified with the Orlicz space $\operatorname{\mathbf{L}}^{{G^{\ast}}}$ generated by conjugate function $G^{\ast}$. On the other hand, $(G^{\ast})^{\ast}=G$ and $(\operatorname{\mathbf{L}}^{{G}})^{\ast}\simeq\operatorname{\mathbf{L}}^{{G^{\ast}}}$ implies reflexivity as well.

Lemma 3.25**.**

Every $v\in\operatorname{\mathbf{L}}^{{G^{\ast}}}$ can be identified with the following functional $F_{v}\in(\operatorname{\mathbf{L}}^{{G}})^{\ast}$:

[TABLE]

Moreover $\|F_{v}\|\leq 2\|v\|_{\operatorname{\mathbf{L}}^{{G^{\ast}}}}$.

Proof.

It is easy to see that $F_{v}$ is linear. By the Hölder inequality we get

[TABLE]

Thus $F_{v}$ is bounded and $\|F_{v}\|\leq 2\|v\|_{\operatorname{\mathbf{L}}^{{G^{\ast}}}}$. ∎

Lemma 3.26** (cf. [10, 11]).**

If $v\in\operatorname{\mathbf{L}}^{{1}}(I,\mathbb{R}^{n})$ is such that for each piecewise constant function $u\in\operatorname{\mathbf{L}}^{{G}}$ satisfy

[TABLE]

then $v\in\operatorname{\mathbf{L}}^{{G^{\ast}}}$ and $\|v\|_{\operatorname{\mathbf{L}}^{{G^{\ast}}}}\leq M$.

Proof.

Define an approximation

[TABLE]

Set $v_{n}=\sum_{i=1}^{n}v_{n,i}\chi_{E_{i}}$. Let $u\in\operatorname{\mathbf{L}}^{{G}}$ be a simple function, define approximation $u_{n}$ of $u$ in the same way. By Jensen inequality

[TABLE]

Hence $\|u_{n}\|_{\operatorname{\mathbf{L}}^{{G}}}\leq\|u\|_{\operatorname{\mathbf{L}}^{{G}}}$. A direct computation yields

[TABLE]

We can find for each $v_{n,i}$ a $z_{n,i}\in\mathbb{R}^{n}$ such that $\langle z_{n,i},v_{n,i}/M\rangle=G(z_{n,i})+G^{\ast}(v_{n,i}/M).$ Suppose that $\sum_{i=1}^{n}\mu(E_{i})G(z_{n,i})>1$. Then there exists $\beta<1$ such that $\sum_{i=1}^{n}\mu(E_{i})G(\beta z_{n,i})=1$. Putting

[TABLE]

we obtain that $\int_{I}G(u)\,dt\leq 1$ and $\|u\|_{\operatorname{\mathbf{L}}^{{G}}}\leq 1$. Therefore

[TABLE]

Now assume that $\mu(E_{i})\sum G(z_{n,i})\leq 1$ and repeat the same computation with $\beta=1$ and obtain

[TABLE]

In both cases we get

[TABLE]

Since $v_{n}\to v$ a.e. we can conclude that $G^{\ast}(v/M)\leq\lim G^{\ast}(v_{n}/M)$. By the Fatou theorem we get

[TABLE]

∎

Lemma 3.27** (cf. [3, 11]).**

For every $F\in(\operatorname{\mathbf{L}}^{{G}})^{\ast}$ there exists unique $v\in\operatorname{\mathbf{L}}^{{G^{\ast}}}$ such that for every $u\in\operatorname{\mathbf{L}}^{{G}}$

[TABLE]

Proof.

For a measurable subset $E\subset I$ define $\chi^{N}_{E}(x)=(\chi_{E},\dots,\chi_{E})$. Note that $\chi^{N}_{E}\in\operatorname{\mathbf{L}}^{{G}}$. Set

[TABLE]

For every sequence $\{E_{i}\}$ of measurable and pairwise disjoint subsets of $I$ such that $E=\bigcup E_{i}$ we have $\chi^{N}_{E}=\sum\chi^{N}_{E_{i}}$ and

[TABLE]

Suppose that there exists a sequence $\{E_{i}\}$ of measurable sets and $\delta>0$ such that $\mu(E_{i})\to 0$ and $\|\chi^{N}_{E_{i}}\|_{\operatorname{\mathbf{L}}^{{G}}}>\delta$ for all $i$. Then

[TABLE]

A contradiction. From inequality

[TABLE]

we obtain that if $\mu(E_{i})\to 0$ then $|\phi(E_{i})|\to 0$. Thus a set function $\phi$ is $\sigma$-additive and absolutely continuous with respect to Lebesgue measure.

It follows from the Radon-Nikodym theorem that there exists a function $v\in\operatorname{\mathbf{L}}^{{1}}(I,\mathbb{R}^{N})$ such that

[TABLE]

For every step function $u=\sum c_{i}\chi_{E_{i}}$, by linearity of $F$,

[TABLE]

By lemma 3.26 we get that $v\in\operatorname{\mathbf{L}}^{{G^{\ast}}}$. Assume now that $u$ is bounded. Choose a sequence of step functions $\{u_{n}\}$ such that

[TABLE]

where $E_{i}$ are disjoint and

[TABLE]

Clearly, $u_{n}\to u$ a.e. and the sequence $\{u_{n}\}$ is uniformly bounded. It follows that

[TABLE]

Suppose that $u$ is an arbitrary function in $\operatorname{\mathbf{L}}^{{G}}$. There exists a sequence $\{u_{n}\}$ of bounded functions which converges a.e. to $u$ such that $|u_{n}(t)|\leq|u(t)|$ a.e. Thus

[TABLE]

It remains to show that $v$ is unique. Suppose that $v_{1}$ and $v_{2}$ represent $F$. Then we have

[TABLE]

for all $u\in\operatorname{\mathbf{L}}^{{\infty}}$. Thus $v_{1}=v_{2}$. ∎

As a consequence we obtain that $\operatorname{\mathbf{L}}^{{G^{\ast}}}\simeq(\operatorname{\mathbf{L}}^{{G}})^{\ast}$. Since $G^{\ast\ast}=G$, we also get $\operatorname{\mathbf{L}}^{{G}}\simeq(\operatorname{\mathbf{L}}^{{G^{\ast}}})^{\ast}$.

Remark 3.28**.**

If G-function does not satisfies $\Delta_{2}$ condition then $\operatorname{\mathbf{L}}^{{G}}$ is not reflexive and $(\operatorname{\mathbf{L}}^{{G}})^{\ast}$ is not isomorphic to $\operatorname{\mathbf{L}}^{{G^{\ast}}}$ (see [3, 11]).

4. Orlicz-Sobolev spaces

The Orlicz-Sobolev space ${\operatorname{\mathbf{W}}^{1}}\operatorname{\mathbf{L}}^{{G}}={\operatorname{\mathbf{W}}^{1}}\operatorname{\mathbf{L}}^{{G}}(I,\mathbb{R}^{n})$ is defined to be

[TABLE]

For $u\in{\operatorname{\mathbf{W}}^{1}}\operatorname{\mathbf{L}}^{{G}}$ we define

[TABLE]

Define ${\operatorname{\mathbf{W}}^{1}_{0}}\operatorname{\mathbf{L}}^{{G}}={\operatorname{\mathbf{W}}^{1}_{0}}\operatorname{\mathbf{L}}^{{G}}(I,\mathbb{R}^{n})$ as the closure of $C^{1}_{0}(I,\mathbb{R}^{n})$ in ${\operatorname{\mathbf{W}}^{1}}\operatorname{\mathbf{L}}^{{G}}$ with respect to the $\|\cdot\|_{{\operatorname{\mathbf{W}}^{1}}\operatorname{\mathbf{L}}^{{G}}}$.

Theorem 4.1**.**

The space $({\operatorname{\mathbf{W}}^{1}}\operatorname{\mathbf{L}}^{{G}},\|\cdot\|_{{\operatorname{\mathbf{W}}^{1}}\operatorname{\mathbf{L}}^{{G}}})$ is a separable reflexive Banach space.

Proof is standard and will be omitted, see for instance [26]. If $G(x)=\frac{1}{p}|x|^{p}$, then the Orlicz-Sobolev space ${\operatorname{\mathbf{W}}^{1}}\operatorname{\mathbf{L}}^{{G}}$ coincides with the Sobolev space $\operatorname{\mathbf{W}}^{1,p}(I,\mathbb{R}^{n})$. Observe that $u_{n}\to u$ in ${\operatorname{\mathbf{W}}^{1}}\operatorname{\mathbf{L}}^{{G}}$ is equivalent to $R_{G}(u_{n}-u)\to 0$ and $R_{G}(\dot{u}_{n}-\dot{u})\to 0$.

On ${\operatorname{\mathbf{W}}^{1}}\operatorname{\mathbf{L}}^{{G}}$ one can introduce another norm (cf. [27]):

[TABLE]

Proposition 4.2**.**

A function $\|\cdot\|_{1,{\operatorname{\mathbf{W}}^{1}}\operatorname{\mathbf{L}}^{{G}}}$ is an equivalent norm on ${\operatorname{\mathbf{W}}^{1}}\operatorname{\mathbf{L}}^{{G}}$. Moreover

[TABLE]

Proof.

The proof that $\|\cdot\|_{1,{\operatorname{\mathbf{W}}^{1}}\operatorname{\mathbf{L}}^{{G}}}$ is a norm is similar to the proof of Theorem 3.2 and is left to the reader. For the other part, note that

[TABLE]

implies

[TABLE]

From this $\|u\|_{\operatorname{\mathbf{L}}^{{G}}}\leq\|u\|_{1,{\operatorname{\mathbf{W}}^{1}}\operatorname{\mathbf{L}}^{{G}}}$ and $\|\dot{u}\|_{\operatorname{\mathbf{L}}^{{G}}}\leq\|\dot{u}\|_{1,{\operatorname{\mathbf{W}}^{1}}\operatorname{\mathbf{L}}^{{G}}}$ and finally, $\|u\|_{{\operatorname{\mathbf{W}}^{1}}\operatorname{\mathbf{L}}^{{G}}}\leq 2\|u\|_{1,{\operatorname{\mathbf{W}}^{1}}\operatorname{\mathbf{L}}^{{G}}}$. Let $\alpha=\max\{\|u\|_{\operatorname{\mathbf{L}}^{{G}}},\|\dot{u}\|_{\operatorname{\mathbf{L}}^{{G}}}\}$. Since $\|u\|_{\operatorname{\mathbf{L}}^{{G}}},\|\dot{u}\|_{\operatorname{\mathbf{L}}^{{G}}}\leq\alpha$,

[TABLE]

and

[TABLE]

Using the above relations, we obtain

[TABLE]

This implies $\|u\|_{1,{\operatorname{\mathbf{W}}^{1}}\operatorname{\mathbf{L}}^{{G}}}\leq 2\alpha\leq 2\|u\|_{{\operatorname{\mathbf{W}}^{1}}\operatorname{\mathbf{L}}^{{G}}}$ ∎

Since there exist $p,q\in(1,\infty)$ such that $\operatorname{\mathbf{L}}^{{q}}\hookrightarrow\operatorname{\mathbf{L}}^{{G}}\hookrightarrow\operatorname{\mathbf{L}}^{{p}}$, the following continuous embeddings exist

[TABLE]

Using standard results from the theory of Sobolev spaces we get

1. (1)

${\operatorname{\mathbf{W}}^{1}}\operatorname{\mathbf{L}}^{{G}}(I,\mathbb{R}^{n})\hookrightarrow\hookrightarrow\operatorname{\mathbf{W}}^{1,1}$ 2. (2)

${\operatorname{\mathbf{W}}^{1}}\operatorname{\mathbf{L}}^{{G}}(I,\mathbb{R}^{n})\hookrightarrow\hookrightarrow\operatorname{\mathbf{L}}^{{q}}$, for all $1\leq q\leq\infty$ 3. (3)

${\operatorname{\mathbf{W}}^{1}}\operatorname{\mathbf{L}}^{{G}}(I,\mathbb{R}^{n})\hookrightarrow\hookrightarrow C(\overline{I})$

As a consequence we have

Theorem 4.3**.**

A function $u\in W^{1}L^{G}$ is absolutely continuous. Precisely, there exist absolutely continuous representative of $u$ such that for all $a,b\in I$

[TABLE]

Directly from definition of ${\operatorname{\mathbf{W}}^{1}_{0}}\operatorname{\mathbf{L}}^{{G}}$ we obtain important property of functions in ${\operatorname{\mathbf{W}}^{1}_{0}}\operatorname{\mathbf{L}}^{{G}}$.

Theorem 4.4**.**

If $u\in{\operatorname{\mathbf{W}}^{1}_{0}}\operatorname{\mathbf{L}}^{{G}}$ then $u=0$ on $\partial I$.

Using embeddings mentioned above we have for every $u\in{\operatorname{\mathbf{W}}^{1}}\operatorname{\mathbf{L}}^{{G}}$

[TABLE]

Theorem 4.5** (Sobolev inequality).**

For every function $u\in{\operatorname{\mathbf{W}}^{1}}\operatorname{\mathbf{L}}^{{G}}$

[TABLE]

where $u_{I}=\frac{1}{\mu(I)}\int_{I}u$.

Proof.

Since $u$ is absolutely continuous, there exists $t_{0}\in I$ such that $u(t_{0})=\frac{1}{\mu(I)}\int_{I}u$ and for every $t\in I$ we have

[TABLE]

By Jensen’s inequality,

[TABLE]

Integrating both sides over $I$ we get

[TABLE]

Thus $\|u-u_{I}\|_{\operatorname{\mathbf{L}}^{{G}}}\leq\mu(I)\|\dot{u}\|_{\operatorname{\mathbf{L}}^{{G}}}$ ∎

In similar way we get

Theorem 4.6** (Poincare inequality).**

For every $u\in{\operatorname{\mathbf{W}}^{1}_{0}}\operatorname{\mathbf{L}}^{{G}}$

[TABLE]

It follows that one can introduce equivalent norm in ${\operatorname{\mathbf{W}}^{1}_{0}}\operatorname{\mathbf{L}}^{{G}}$:

[TABLE]

Every linear functional $F$ on ${\operatorname{\mathbf{W}}^{1}_{0}}\operatorname{\mathbf{L}}^{{G}}$ can be represented in the form

[TABLE]

Where $v_{0},v_{1}\in\operatorname{\mathbf{L}}^{{G^{\ast}}}$. Moreover,

[TABLE]

In the case of Sobolev space $\operatorname{\mathbf{W}}^{1,p}$ the proof is given in [26, proposition 8.14], but it remains the same for Orlicz-Sobolev spaces. As was pointed out in [26], the first assertion of the above proposition holds for every linear functional on ${\operatorname{\mathbf{W}}^{1}}\operatorname{\mathbf{L}}^{{G}}$.

5. Variational setting

In this section we examine the principal part

[TABLE]

of the variational functional associated with Euler-Lagrange equation

[TABLE]

where $u\colon I\to\mathbb{R}^{N}$ and the Lagrangian $L\colon I\times\mathbb{R}^{N}\times\mathbb{R}^{N}\to\mathbb{R}$ is given by $L(t,x,v)=F(t,x,v)+V(t,x)$.

In definition of the Orlicz space we need not to assume that $G$ is differentiable, but when we consider the functional $\operatorname{\mathcal{I}}$ we need it to show that $\operatorname{\mathcal{I}}\in C^{1}$. Throughout this section we will assume, in addition to ($G_{1}$)–($G_{6}$), that $G$ satisfies

1. (7)

$G$ is of a class $C^{1}.$

Remark 5.1**.**

Differentiability of $f$ is not sufficient to differentiability of $f^{\ast}$. But if $f$ is finite, strictly convex, 1-coercive and differentiable then so is $f^{\ast}$. This result is in close relation with Legendre duality (see [21, p. 239] and [1] for more details).

It is well known that if $G$ is continuously differentiable then for all $x,y\in\mathbb{R}^{n}$

[TABLE]

and

[TABLE]

Let $y=x$ in (5). Then $\langle\nabla G(x),x\rangle\leq G(2x)-G(x)$. Therefore, for all $x\in\mathbb{R}^{N}$

[TABLE]

Directly from the above we get

Proposition 5.2**.**

If $u\in\operatorname{\mathbf{L}}^{{G}}$ then $\nabla G(u)\in\operatorname{\mathbf{L}}^{{G^{\ast}}}$.

Lemma 5.3** (cf. [16, lemma A.5]).**

If $u_{n}\to u$ in $\operatorname{\mathbf{L}}^{{G}}$ then $R_{G^{\ast}}(\nabla G(u_{n}))\to R_{G^{\ast}}(\nabla G(u)).$

Proof.

There exists a subsequence $\{u_{n_{k}}\}$ such that $u_{n_{k}}\to u$ a.e., $G(u_{n_{k}})\to G(u)$ a.e. and $G(u_{n_{k}})\leq h\in\operatorname{\mathbf{L}}^{{1}}(I,\mathbb{R})$. By continuity of $\nabla G$ and $G^{\ast}$ we have $\nabla G(u_{n_{k}})\to\nabla G(u)$ a.e. and

[TABLE]

Since $G^{\ast}(\nabla G(x))\leq G(2x)$,

[TABLE]

By dominated convergence theorem $R_{G^{\ast}}(\nabla G(u_{n_{k}}))\to R_{G^{\ast}}(\nabla G(u))$. Since this holds for any subsequence of $\{u_{n}\}$ we have that

[TABLE]

∎

As a direct consequence of the above lemma and Lemma 3.13 we obtain

Proposition 5.4**.**

[TABLE]

5.1. Case I

We shall first examine a special case $F(t,x,v)=G(v)$, now functional (4) takes the form

[TABLE]

Theorem 5.5**.**

$\operatorname{\mathcal{I}}\in C^{1}({\operatorname{\mathbf{W}}^{1}}\operatorname{\mathbf{L}}^{{G}},\mathbb{R})$. Moreover

[TABLE]

Proof.

The proof follows similar lines as [2, th. 3.2] (see also [1, thm 1.4]). First, note that $\dot{u}\in\operatorname{\mathbf{L}}^{{G}}$ implies

[TABLE]

It suffices to show that $\operatorname{\mathcal{I}}$ has at every point $u$ directional derivative $\operatorname{\mathcal{I}}^{\prime}(u)\in({\operatorname{\mathbf{W}}^{1}}\operatorname{\mathbf{L}}^{{G}})^{\ast}$ given by (6) and that the mapping $\operatorname{\mathcal{I}}^{\prime}:{\operatorname{\mathbf{W}}^{1}}\operatorname{\mathbf{L}}^{{G}}\to({\operatorname{\mathbf{W}}^{1}}\operatorname{\mathbf{L}}^{{G}})^{\ast}$, is continuous.

Let $u\in{\operatorname{\mathbf{W}}^{1}}\operatorname{\mathbf{L}}^{{G}}$, $\varphi\in{\operatorname{\mathbf{W}}^{1}}\operatorname{\mathbf{L}}^{{G}}\setminus\{0\}$, $t\in I$, $s\in[-1,1]$ . Define

[TABLE]

By (5) we obtain

[TABLE]

Consequently, $\operatorname{\mathcal{I}}$ has a directional derivative and

[TABLE]

By Lemma 5.2 and Hölder inequality

[TABLE]

To finish the proof it suffices to show that if $u_{n}\to u$ in ${\operatorname{\mathbf{W}}^{1}}\operatorname{\mathbf{L}}^{{G}}$, then $\operatorname{\mathcal{I}}^{\prime}(u_{n})\to\operatorname{\mathcal{I}}^{\prime}(u)$ in $(W^{1}L^{G})^{\ast}$. Using Hölder inequality and Proposition 5.4 we obtain

[TABLE]

∎

5.2. Case II

We turn to general case. Suppose that $F\colon I\times\mathbb{R}^{N}\times\mathbb{R}^{N}\to\mathbb{R}$ satisfies

1. ($F_{1}$)

$F\in C^{1}$ 2. ($F_{2}$)

$|F(t,x,v)|\leq a(|x|)(b(t)+G(v)),$ 3. ($F_{3}$)

$|F_{x}(t,x,v)|\leq a(|x|)(b(t)+G(v)),$ 4. ($F_{4}$)

$G^{\ast}(F_{v}(t,x,v))\leq a(|x|)(c(t)+G^{\ast}(\nabla G(v))).$

where $a\in C(\mathbb{R}_{+},\mathbb{R}_{+})$, $b,c\in\operatorname{\mathbf{L}}^{{1}}(I,\mathbb{R}_{+}).$

If $G(v)=|v|^{p}$ then conditions ($F_{2}$), ($F_{3}$) and ($F_{4}$) take the standard form (Theorem 1.4 from [1]). In [2] there are similar conditions with $G(v)=\Phi(|v|)$, where $\Phi$ is an N-function. In this case, condition ($F_{4}$) takes the form $|F_{v}(t,x,v)|\leq\tilde{a}(|x|)(\tilde{c}(t)+\Phi^{\prime}(|u|))$. In anisotropic case we need to use $G^{\ast}$, because vector valued G-function is not necessarily monotone with respect to $|\cdot|$.

Directly from ($F_{3}$), ($F_{4}$) and Proposition 5.2 we have

Lemma 5.6**.**

If $u\in{\operatorname{\mathbf{W}}^{1}}\operatorname{\mathbf{L}}^{{G}}$, then $F_{x}(\cdot,u,\dot{u})\in\operatorname{\mathbf{L}}^{{1}}$ and $F_{v}(\cdot,u,\dot{u})\in\operatorname{\mathbf{L}}^{{G^{\ast}}}$.

Proof.

Define non decreasing function

[TABLE]

Then, for $u\in{\operatorname{\mathbf{W}}^{1}}\operatorname{\mathbf{L}}^{{G}}$ we have

[TABLE]

Let $u\in{\operatorname{\mathbf{W}}^{1}}\operatorname{\mathbf{L}}^{{G}}$. By (7) and ($F_{3}$)

[TABLE]

Moreover, by Proposition 5.2 and ($F_{4}$)

[TABLE]

∎

Theorem 5.7**.**

$\operatorname{\mathcal{I}}\in C^{1}({\operatorname{\mathbf{W}}^{1}}\operatorname{\mathbf{L}}^{{G}},\mathbb{R})$. Moreover

[TABLE]

Proof.

By ($F_{2}$),

[TABLE]

It suffices to show that directional derivative $\operatorname{\mathcal{I}}^{\prime}(u)\in({\operatorname{\mathbf{W}}^{1}}\operatorname{\mathbf{L}}^{{G}})^{\ast}$ exists, is given by (8) and that the mapping $\operatorname{\mathcal{I}}^{\prime}:{\operatorname{\mathbf{W}}^{1}}\operatorname{\mathbf{L}}^{{G}}\to({\operatorname{\mathbf{W}}^{1}}\operatorname{\mathbf{L}}^{{G}})^{\ast}$ is continuous.

Let $u\in{\operatorname{\mathbf{W}}^{1}}\operatorname{\mathbf{L}}^{{G}}$, $\varphi\in{\operatorname{\mathbf{W}}^{1}}\operatorname{\mathbf{L}}^{{G}}\setminus\{0\}$, $t\in I$, $s\in[-1,1]$. Define

[TABLE]

By ($F_{3}$), continuity of $\varphi$, (7) and the fact that $u+s\varphi\in{\operatorname{\mathbf{W}}^{1}}\operatorname{\mathbf{L}}^{{G}}$ we obtain

[TABLE]

By the Fenchel inequality, ($F_{4}$) and Lemma 5.6 we obtain

[TABLE]

It follows that

[TABLE]

Consequently, $\operatorname{\mathcal{I}}$ has a directional derivative and

[TABLE]

By Lemma 5.6, the Hölder inequality and (3) we get

[TABLE]

To finish the proof it suffices to show that $\operatorname{\mathcal{I}}^{\prime}$ is continuous. Since $u_{n}\to u$ in ${\operatorname{\mathbf{W}}^{1}}\operatorname{\mathbf{L}}^{{G}}$, it follows that $u_{n}\to u$ in $\operatorname{\mathbf{L}}^{{G}}$, $\dot{u}_{n}\to\dot{u}$ in $\operatorname{\mathbf{L}}^{{G}}$ and there exists $M>0$ such that $\|u_{n}\|_{{\operatorname{\mathbf{W}}^{1}}\operatorname{\mathbf{L}}^{{G}}}<M$.

By Lemma 3.12 we have $G(\dot{u}_{n})\to G(\dot{u})$ in $\operatorname{\mathbf{L}}^{{1}}(I,\mathbb{R})$. Hence there exists a subsequence $\{u_{n_{k}}\}$ and $h\in\operatorname{\mathbf{L}}^{{1}}(I,\mathbb{R})$ such that

[TABLE]

By ($F_{3}$) and since $\{u_{n_{k}}\}$ is bounded, we obtain

[TABLE]

By ($F_{1}$) we have

[TABLE]

for a.e $t\in I$. Applying Lebesgue Dominated Convergence Theorem we obtain

[TABLE]

Since this holds for any subsequence of $\{u_{n}\}$ we have that

[TABLE]

By ($F_{4}$) and Lemma 5.6

[TABLE]

In the same way as in the proof of Lemma 5.3 we obtain

[TABLE]

By continuity of $F_{v}$ we obtain

[TABLE]

for a.e $t\in I$ and consequently

[TABLE]

It follows that

[TABLE]

Application of Lemma 3.13 to $R_{G^{\ast}}$ yields $\|F_{v}(\cdot,u_{n},\dot{u_{n}})-F_{v}(\cdot,u,\dot{u})\|_{\operatorname{\mathbf{L}}^{{G^{\ast}}}}\to 0$. By Hölder inequality

[TABLE]

Finally,

[TABLE]

∎