### Abstract

The article deals with the existence of solutions of some unilateral problems in the Orlicz-Sobolev spaces framework when the right-hand side is a Radon measure.

**Mathematics Subject Classification**: 35K86.

##### Keywords:

unilateral problem; radon measure; Orlicz-Sobolev spaces### 1 Introduction

We deal with boundary value problems

where

*T *> 0 and Ω is a bounded domain of **R**^{N}, with the segment property. *a *: Ω × **R **× **R**^{N }→ **R**^{N }is a Carathéodory function (that is, measurable with respect to *x *in Ω for every (*t, s, ξ*) in **R **× **R **× **R**^{N }, and continuous with respect to (*s, ξ*) in **R **× **R**^{N }for almost every *x *in Ω) such that for all *ξ, ξ** ∈ **R**^{N }, *ξ *≠ *ξ**,

where *c *(*x*,*t*) belongs to
*P *is an *N*-function such that *P *≪ *M *and *k*_{i }(*i *= 1,2,3,4) belongs to **R**^{+ }and *α *to

There have obviously been many previous studies on nonlinear differential equations with nonsmooth coefficients and measures as data. The special case was cited in the references (see [1,2]).

It is noteworthy that the articles mentioned above differ in significant way, in the
terms of the structure of the equations and data. In [1], when *f *∈ *L*^{1}(0,*T*;*L*^{1}(Ω)) and *u*_{0 }∈ *L*^{1}(Ω). The authors have shown the existence of solutions *u *of the corresponding equation of the problem
*q *such that

In this article, we are interested with an obstacle parabolic problem with measure as data. We give an improved regularity result of the study of Boccardo et al. [1].

In [1], the authors have shown the existence of a weak solutions for the corresponding equation,
the function *a*(*x, t, s, ξ*) was assumed to satisfy a polynomial growth condition with respect to *u *and ∇*u*. When trying to relax this restriction on the function *a*(., s, *ξ*), we are led to replace the space *L*^{p}(0, *T*; *W*^{1,p}(*Ω*)) by an inhomogeneous Sobolev space *W*^{1,x}*L*_{M }built from an Orlicz space *L*_{M }instead of *L*^{p}, where the *N*-function *M *which defines *L*_{M }is related to the actual growth of the Carathéodory's function.

For simplicity, one can suppose that there exist *α *> 0, *β *> 0 such that

### 2 Preliminaries

Let *M *: **R**^{+ }→ **R**^{+ }be an *N*-function, i.e. *M *is continuous, convex, with *M*(*t*) > 0 for
*t *→ 0 and
*t *→ ∞. Equivalently, *M *admits the representation:
*a *: **R**^{+ }→ **R**^{+ }is non-decreasing, right continuous, with *a*(0) = 0, *a*(*t*) > 0 for *t *> 0 and *a*(*t*) → ∞ as *t *→ ∞. The *N*-function
*M *is defined by

The *N*-function *M *is said to satisfy the Δ_{2 }condition if, for some *k *> 0:

when this inequality holds only for *t *≥ *t*_{0 }> 0, *M *is said to satisfy the Δ_{2 }condition near infinity.

Let *P *and *Q *be two *N*-functions. *P *≪ *Q *means that *P *grows essentially less rapidly than *Q*; i.e., for each *ε *> 0,

Let Ω be an open subset of **R**^{N}. The Orlicz class
*L*_{m}(Ω)) is defined as the set of (equivalence classes of) real-valued measurable functions
*u *on Ω such that

Note that *L*_{M}(Ω) is a Banach space under the norm
*L*_{M}(Ω). The closure in *L*_{M}(Ω) of the set of bounded measurable functions with compact support in
*E*_{M}(Ω). The equality *E*_{M}(Ω) = *L*_{M}(Ω) holds if and only if *M *satisfies the Δ_{2 }condition, for all *t *or for *t *large according to whether Ω has infinite measure or not.

The dual of *E*_{M}(Ω) can be identified with
*L*_{M}(Ω) is reflexive if and only if *M *and
_{2 }condition, for all *t *or for *t *large, according to whether Ω has infinite measure or not.

We now turn to the Orlicz-Sobolev space. *W*^{1}*L*_{M}(Ω) (resp. *W*^{1}*E*_{M}(Ω)) is the space of all functions *u *such that *u *and its distributional derivatives up to order 1 lie in *L*_{M}(Ω) (resp. *E*_{M}(Ω)). This is a Banach space under the norm
*W*^{1}*L*_{M}(Ω) and *W*^{1}*E*_{M}(Ω) can be identified with subspaces of the product of *N *+ 1 copies of *L*_{M}(Ω). Denoting this product by Π*L*_{M}, we will use the weak topologies
*W*^{1}*E*_{M}(Ω) and the space
*W*^{1}*L*_{M}(Ω). We say that *u*_{n }converges to *u *for the modular convergence in *W*^{1}*L*_{M}(Ω) if for some
*α*| ≤ 1. This implies convergence for
*M *satisfies the Δ_{2 }condition on **R**^{+}(near infinity only when Ω has finite measure), then modular convergence coincides
with norm convergence.

Let

If the open set Ω has the segment property, then the space

For *k *> 0, *s *∈ **R**, we define the truncation at height *k*,*T*_{k}(*s*) = [*k *- (*k *- |*s*|)_{+}]*sign*(*s*).

The following abstract lemmas will be applied to the truncation operators.

**Lemma 2.1 **[7]*Let F *: **R **→ **R ***be uniformly lipschitzian, with F*(0) = 0. *Let M be an N-function and let *
*(resp*.

*Then *
*(resp*.
*Moreover, if the set of discontinuity points of F' is finite, then*

Let Ω be a bounded open subset of **R**^{N}, *T *> 0 and set *Q *= Ω × ]0, *T*[. Let *m *≥ 1 be an integer and let *M *be an *N*-function. For each *α *∈ **IN**^{N }, denote by
*Q *of order *α *with respect to the variable *x *∈ **R**^{N}. The inhomogeneous Orlicz-Sobolev spaces are defined as follows

The last space is a subspace of the first one, and both are Banach spaces under the
norm
*L*_{m}(*Q*) which have as many copies as there are *α*-order derivatives, |*α*| ≤ *m*. We shall also consider the weak topologies
*u *∈ *W*^{m, x}*L*_{M}(*Q*), then the function : *t *↦ *u*(*t*) = *u*(*t*,.) is defined on [0, *T*] with values in *W*^{m}*L*_{M}(Ω). If, further, *u *∈ *W*^{m,x}*E*_{M}(*Q*) then the concerned function is a *W*^{m}*E*_{M}(Ω)-valued and is strongly measurable. Furthermore, the following imbedding holds:
*W*^{m,x}*E*_{M}(*Q*) ⊂ *L*^{1}(0,*T*; *W*^{m}*E*_{M}(Ω)). The space *W*^{m,x}*L*_{M}(*Q*) is not in general separable, if *u *∈ *W*^{m,x}*L*_{M}(*Q*), we cannot conclude that the function *u*(*t*) is measurable on [0,*T*]. However, the scalar function *t *↦ ||*u*(*t*)||_{M,Ω}, is in *L*^{1}(0,*T*). The space
*W*^{m,x}*E*_{M}(*Q*) of
*u *of the closure of
*W*^{m,x}*L*_{M}(*Q*), of some subsequence
*m*,

this implies that (*u*_{i}) converges to *u *in *W*^{m,x}*L*_{M}(*Q*) for the weak topology

Furthermore,
*C *> 0 such that for all

*F *being the dual space of
*F*_{0 }is then given by

We can easily check, using Lemma 4.4 of [6], that each uniformly lipschitzian mapping *F*, with *F*(0) = 0, acts in inhomogeneous Orlicz-Sobolev spaces of order 1: *W*^{1,x}*L*_{M}(*Q*) and

### 3 Main results

First, we give the following results which will be used in our main result.

#### 3.1 Useful results

Hereafter, we denote by
**R**^{N}, and for a fixed *t *∈ [0, *T*], we denote *μ*(*θ*) = *meas*{(*x*,*t*) : |*u*(*x, t*)| > *θ*}.

**Lemma 3.1 **[8]*Let *
*and let fixed t *∈ [0, *T*], *then we have*

*and where *
*is defined by*

**Lemma 3.2 ***Under the hypotheses (1.1)-(1.3), if f, u*_{0 }*are regular functions and f, u*_{0 }≥ 0, *then there exists at least one positive weak solution of the problem*

*such that*

**Proof**

Let *u *be a continuous function, we say that *u *satisfies (*) if: there exists a continuous and increasing function *β *such that ||*u*(*t*) - *u*(*s*)||_{2 }≤ *β*(||*u*_{0}||_{2})|*t *- *s*|, where *u*_{0}(*x*) = *u*(*x*, 0).

Let

Let us consider the set
*C *is a closed convex of

We claim that the problem

has a weak solution which is unique in the sense defined in [9].

Indeed, let us consider the approximate problem

where the functional Φ is defined by Φ : *X *→ **R **∪ {+ ∞} such that

The existence of such *u*_{n }∈ *X *was ensured by Kacur et al. [10].

Following the same proof as in [9], we can prove the existence of a solution *u *of the problem (*E'*) as limit of *u*_{n }(for more details see [9]).

**Lemma 3.3 ***Let *
*such that *
*and *

*Then, there exists a smooth function *(*v*_{j}) *such that*

*v*_{j }→ *v for the modular convergence in *

*for the modular convergence in *

For the proof, we use the same technique as in [11] in the parabolic case.

#### 3.2 Existence result

Let *M *be a fixed *N*-function, we define *K *as the set of *N*-function *D *satisfying the following conditions:

*i*) *M*(*D*^{-1}(*s*)) is a convex function,

*ii*)

*iii*) There exists an *N*-function *H *such that

**Theorem 3.1 ***Under the hypotheses (1.1)-(1.5), The problem *(*P*) *has at least one solution u in the following sense:*

*for all φ *∈ *D*(**R**^{N+1}) *which are zero in a neighborhood of *(0, *T*) × ∂ Ω *and *{*T*} × Ω.

**Remark 3.1 ***(1) If ψ *= - ∞ *in the problem (P), then the above theorem gives the same regularity as in the elliptic
case.*

*(2) An improved regularity is reached for all *N-*function satisfying the conditions (i)*-*(ii)*-*(iii)*.

*For example*,
*for all *

In the sequel and throughout the article, we will omit for simplicity the dependence
on *x *and *t *in the function *a*(*x, t, s, ξ*) and denote ϵ(*n, j, μ, s, m*) all quantities (possibly different) such that

and this will be in the order in which the parameters we use will tend to infinity,
that is, first *n*, then *j, μ, s*, and finally *m*. Similarly, we will write only ϵ(*n*), or ϵ(*n, j*),... to mean that the limits are made only on the specified parameters.

#### 3.2.1 A sequence of approximating problems

Let (*f*_{n}) be a sequence in *D*(*Q*) which is bounded in *L*^{1}(*Q*) and converge to *μ *in *M*_{b}(*Q*).

Let
*D*(Ω) which is bounded in *L*^{1}(Ω) and converge to *u*_{0 }in *M*_{b}(Ω).

We define the following problems approximating the original (*P*):

**Lemma 3.4 ***Under the hypotheses (1.1)-(1.3), there exists at least one solution u*_{n }*of the problem *(*P*_{n}) *such that *
*a*.*e*. *in Q*.

For the proof see Lemma 3.2.

#### 3.2.2 *A priori *estimates

**Lemma 3.5 ***There exists a subsequence of *(*u*_{n}) *(also denoted *(*u*_{n})*), there exists a measurable function u such that:*

Proof:

Recall that *u*_{n }≥ 0 since *f*_{n }≥ 0.

Let *h *> 0 and consider the following test function *v *= *T*_{h}(*u*_{n }- *T*_{k}(*u*_{n})) in (*P*_{n}), we obtain

We have

So,

Now, let us fix *k *> ||*ψ*||_{∞}, we deduce the fact that:

Let *h *to tend to zero, one has

Let us use as test function in (*P*_{n}),*v *= *T*_{k}(*u*_{n}), then as above, we obtain

Then (*T*_{k}(*u*_{n})_{n}) is bounded in

*T*_{k}(*u*_{n}) ⇀ *ω*_{k}, weakly in
*E*_{M}(*Q*) and a.e in *Q*.

Let consider the *C*^{2 }function defined by

Multiplying the approximating equation by
*η*_{k}(*u*_{n}) being bounded in
*η*_{k}(*u*_{n}) is compact in *L*^{1 }(*Q*).

Following the same way as in [2], we obtain for every *k *> 0,

Using now the estimation (3.1) and Fatou's lemma to obtain

Let fixed a *t *∈ [0, *T*]. We argue now as for the elliptic case, the problem becomes:

We denote *g*_{n }:= *nT*_{n}((*u*_{n }- *ψ*)^{-}).

Let *φ *be a truncation defined by

for all *θ, h *> 0.

Using *v *= *φ *(*u*_{n}) as a test function in the approximate elliptic problem

here and below *C *denote positive constants not depending on *n*.

By using Lemma 3.1, we obtain (supposing -*μ'*(*θ*) > 0 which does not affect the proof) and following the same way as in [8], we have for *D *∈ *K*,

Let denote

Using Lemma 3.1, denoting

Remark also that

Combining the inequalities (3.5) and (3.6) we obtain,

and since the function

Then, the sequence (*u*_{n}) is bounded in
*N*-function *D *∈ *K*.

#### 3.3 Almost everywhere convergence of the gradients

**Lemma 3.6 ***The subsequence *(*u*_{n}) *obtained in Lemma 3.5 satisfies:*

Proof:

Let *m *> 0, *k *> 0 such that *m *> *k*. Let *ρ*_{m }be a truncation defined by

where *v*_{j }∈ *D*(*Q*) such that *v*_{j }≥ *ψ *and *v*_{j }→ *T*_{k}(*u*) with the modular convergence in

*ω*_{μ }is the mollifier function defined in Landes [13], the function *ω*_{μ,j }have the following properties:

Set *v *= (*T*_{k}(*u*_{n}) - *ω*_{μ,j}) *ρ*_{m}(*u*_{n}) as test function, we have

Let us recall that for
*u*_{nσ }(see [14]) such that

Remark also that,

and it is easy to see that

Concerning *I*_{2},

As in *I*_{1}, we obtain

and

thus by using the fact that

So,

About *I*_{3},

Set Φ(*s*) = *s*^{2}/2, Φ ≥ 0,then

So,

and easily we deduce,

Finally we conclude that:

We are interested now with the terms of (1)-(4).

About (1):

recall that *ρ*_{m}(*u*_{n}) = 1 on {|*u*_{n}| ≤ *k*}.

Let

By using the inequality (1.3), we can deduce the existence of some measurable function
*h*_{k }such that

since

and

Then,

Following the same way as in *J*_{2}, one has

Concerning the terms *J*_{4 }:

Letting *n *→ ∞, then

Taking now the limits *j *→ ∞ and after *μ *→ ∞ in the last equality, we obtain

Then,

About (2):

Since (*u*_{n}) is bounded in
*iii*), we have (*a*(., *u*_{n},∇*u*_{n})) is bounded in *L*_{H}(*Q*), then

so,

About (4):

Since *u *≥ *ψ*, then *T*_{k}(*u*) ≥ *T*_{k}(*ψ*) and there exist a smooth function *v*_{j }≥ *T*_{k}(*ψ*) such that *v*_{j }→ *T*_{k}(*u*) for the modular convergence in

Taking into account now the estimation of (1), (2), (4)and (5), we obtain

On the other hand,

each term of the last right hand side is of the form ϵ(*n, j, s*), which gives