2002-Fez conference on Partial Differential Equations, Electronic Journal of Differential Equations, Conference 09, 2002, pp 1–8. http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp)

On the L^{∞}-regularity of solutions of nonlinear elliptic equations in Orlicz spaces ^{∗}

Lahsen Aharouch, Elhoussine Azroul, & Abdelmoujib Benkirane

Abstract

Our main result is a maximum principle bounding the absolute values of the solution in terms of the supremum of the absolute values of the boundary data.

1 Introduction

Let Ω be a bounded Lipshitz domain in ℝ^{n} (n ≥ 1), let M(t) be an N-function i.e. continuous, convex, with M(t) > 0 for t > 0, M(t)/t → 0 as t → 0 and M(t)/t → ∞ as t → ∞, and m(t) be its right derivatives. Consider the nonlinear boundary-value problem

Au = − div a(∇u) = f in Ω

u = θ in ∂Ω (1.1)

with prescribed boundary datum θ, where a = {a_{i}, 1 ≤ i ≤ n} is a vector of Carathéodory functions defined on ℝ^{n} satisfying the hypotheses

(H1) |a(ξ)| ≤ λ_{1}m(|ξ|) for all ξ ∈ ℝ^{n} and some positive constant λ_{1}.

(H2) a(ξ).ξ ≥ λ_{2}|ξ|m(|ξ|) for all ξ ∈ ℝ^{n} and some positive constant λ_{2}.

Definition 1.1 Let θ ∈ W^{1}L_{M}(Ω) and f ∈ L^{1}(Ω). A function u ∈ W^{1}L_{M}(Ω) is called a weak solution of the boundary-value problem (1.1) if u−θ ∈ W_{0}^{1}L_{M}(Ω) and

∫_{Ω} a(∇u).∇φ = ∫_{Ω} f.φ for all φ ∈ C_{0}^{∞}(Ω).

Here W_{0}^{1}L_{M}(Ω) and W^{1}L_{M}(Ω) denote the Orlicz-Sobolev Spaces associated to the N-function M (see section 2).

∗Mathematics Subject Classifications: 35J60.

Key words: Orlicz Sobolev spaces, boundary-value problems.

\begin{matrix}c \\ ◯ \end{matrix}2002 Southwest Texas State University.

Published December 28, 2002.

2 On the L^{∞}-regularity

Recently, Fuchs and Gongbao proved in [6, Theorem 1.1] that if u is a weak solution of (1.1) with the second member f lies in L^{∞}(Ω), and sup_{∂Ω} θ(x) < ∞, then u is bounded from above; i.e.,

sup_{Ω} u(x) ≤ const(sup_{∂Ω} θ(x), ∥u∥_{L^{1}(Ω)}, n, |Ω|, M, ∥f∥_{L^{∞}Ω)}, λ_{1}, λ_{2}) < ∞.

For this, the authors have supposed additionally to (H1)-(H2) that the Nfunction M satisfies the Δ_{2}-condition near infinity.

The aim of this paper is to prove (Theorem 3.2) the previous statement for the general operators,

Au = − div(a(x, u, ∇u))  (1.2)

without assuming the Δ_{2}-condition. To do this, we replace the hypothesis (H1) by the more general growth condition,

|a(x, s, ξ)| ≤ c(x) + k_{1}\overline{P}^{−1}M(k_{2}|s|) + k_{3}\overline{M}^{−1}M(k_{4}|ξ|)  (1.3)

and the hypothesis (H2) by

a(x, s, ξ)ξ ≥ αM(\frac{|ξ|}{β}).  (1.4)

(see section 3). To generalize theorem of [6], in our case, we need to prove the following approximating result (see theorem 3.2)

W_{0}^{1,1}(Ω) ∩ W^{1}L_{M}(Ω) = W_{0}^{1}L_{M}(Ω).

which guaranties that φ = max(u − k, 0) can be taken as a test function (for details see theorem 3.2).

When M(t) = |t|^{p} (p > 1) (i.e., a satisfies the polynomial growth condition), the regularity result of the solution of (1.1) are investigated in [14] and [11]. Non-standard examples of M(t) which occur in the mechanics of solids and fluids are M(t) = t ln(1 + t), M(t) = ∫_{0}^{t} s^{1−α}(arsinh s)^{α}ds (0 ≤ α ≤ 1) and M(t) = t ln(1 + ln(1 + t)) (see [8, 9, 10, 6]) for more details). When Au = −Δu (corresponding to the Poisson equation), the reader is referred to [3], where the regularity of u is studied in Orlicz Spaces with respect to the second member f (in particular where f is a measure).

Finally. note that some problems of the calculus of variations (see [6, remark 1.2]) can be lead to the equation (3.1) as in section 3, contributions in this sense include the works [5, 2].

2 Preliminaries

2-1 Let M : ℝ^{+} → ℝ^{+} be an N-function, i.e. M is continuous, convex, with M(t) > 0 for t > 0, M(t)/t → 0 as t → 0 and M(t)/t → ∞ as t → ∞.

The N-function conjugate to M is defined as \overline{M}(t) = sup\{st − M(t), s ≥ 0\}. We will extend these N-functions into even functions on all ℝ.

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

M(2t) ≤ kM(t) ∀t ≥ 0.  (2.1)

When this inequality holds only for t ≥ some t_{0} > 0, M is said to satisfy the Δ_{2}-condition near infinity. Moreover, we have the following Young’s inequality

∀s, t ≥ 0, st ≤ M(s) + \overline{M}(t)  (2.2)

Let P and M be two N-functions. P << M means that P grows essentially less rapidly than M, i.e. for each ε > 0, \frac{P (t)}{M(εt)} → 0 as t → ∞ This is the case if and only if

lim_{t→∞} \frac{M^{−1}(t)}{P^{−1}(t)} = 0

2-2 Let Ω be an open subset of ℝ^{n}. The Orlicz class K_{M}(Ω) [resp. The Orlicz space L_{M}(Ω)] is defined as the set of (equivalence classes of) real-valued measurable functions u on Ω such that,

∫_{Ω} M(u(x))dx < +∞ (resp. ∫_{Ω} M(\frac{u(x)}{λ})dx for some λ > 0).

L_{M}(Ω) is a Banach space under the norm

∥u∥_{M} = inf\{λ > 0 : ∫_{Ω} M(\frac{u(x)}{λ})dx ≤ 1\}

and K_{M}(Ω) is a convex subset of L_{M}(Ω). The closure in L_{M}(Ω) of the set of bounded measurable functions with compact support in \overline{Ω} is denoted by E_{M}(Ω). 2-3 We now turn to the Orlicz-Sobolev spaces, W^{1}L_{M}(Ω) [resp. W^{1}E_{M}(Ω)] is the space of functions u such that u and its distributional derivatives up to order 1 lie in L_{M}(Ω) [resp. E_{M}(Ω)]. It is a Banach space under the norm

∥u∥_{1.M} = ∑_{|α|≤1} ∥D^{α}u∥_{M}

Thus, 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 ΠL_{M}, we will use the weak topologies σ(ΠL_{M}, ΠE\overline{M}) and σ(ΠL_{M}, ΠL\overline{M}).

The space W_{0}^{1}E_{M}(Ω) is defined as the (norm) closure of the Schwartz space D(Ω) in W^{1}E_{M}(Ω) and the space W_{0}^{1}L_{M}(Ω) as the σ(ΠL_{M}, ΠE\overline{M}) closure of D(Ω) in W^{1}L_{M}(Ω). Now, we recall the following concept.

Definition 2.1 A domain Ω has the segment property if for every x ∈ ∂Ω there exists an open set G_{x} and a nonzero vector y_{x} such that x ∈ G_{x} and if z ∈ \overline{Ω} ∩ G_{x}, then z + ty_{x} ∈ Ω for all 0 < t < 1.

Recall that if Ω is a bounded Lipshitz domain it satisfies the segment property (see [1]).

3 Main results

Let Ω be a bounded Lipshitz domain in ℝ^{n}. Our first aim of this section is to prove the following result which play an important role in the proof of the regularity result (Theorem 3.2). Note that some ideas of the proof of Theorem 3.1 are inspired from the analogous of Theorem 1.3 of [12].

Theorem 3.1 Let M be an N-function, Ω be a bounded open domain of ℝ^{N} satisfying the segment property. Then

W_{0}^{1,1}(Ω) ∩ W^{1}L_{M}(Ω) = W_{0}^{1}L_{M}(Ω).

Proof. We use the notation

ũ = \begin{cases}u & in Ω \\ 0 & in ℝ^{N}\Ω \end{cases}

Step 1 We show that, W_{0}^{1,1}(Ω) ∩ W^{1}L_{M}(Ω) ⊂ W_{0}^{1}L_{M}(Ω).

Let u ∈ W_{0}^{1,1}(Ω) ∩ W^{1}L_{M}(Ω), set K =: \overline{\{x ∈ Ω, u(x) ≠ 0\}}^{ℝ^{N}} (closure in ℝ^{N}). Then K is a compact in ℝ^{N}, and K ⊂ \overline{Ω}.

If K ⊂ Ω. Let j_{ε} a mollifier function, the convolution j_{ε} ∗ u belongs in C_{0}^{∞}(Ω), for all 0 < ε < dist(K, ∂Ω). By [12, Lemma 6] we get j_{ε} ∗ u → u in W^{1}L_{M}(Ω) for σ(ΠL_{M}, ΠL\overline{M}), as ε → 0^{+}, this proves that u ∈ W_{0}^{1}L_{M}(Ω).

If K ∩ ∂Ω ≠ ∅. For all x ∈ ∂Ω, let G_{x} and y_{x} be respectively the open set and the nonzero vector given by Definition 2.1. Set F = K ∩ (\overline{Ω}\ ∪_{x∈∂Ω} G_{x}), then F is a subset compact of Ω. Hence there exist an open G_{0} such that F ⊂ G_{0} ⊂⊂ Ω. Since K is compact, we can found finitely sets G_{x}; (let us rename them G_{1}, …, G_{k}) such that K ⊂ G_{0} ∪ ⋯ ∪ G_{k}. Moreover, as in the proof of [1, Theorem 3.18], we can construct some open sets G̃_{0}, G̃_{1}, …, G̃_{k} which cover K, such that \overline{G̃_{j}} ⊂ G_{j} for every j. Now, let Θ = \{θ_{j}, 0 ≤ j ≤ k\} be a partition of unity subordinate to {G̃_{j}, 0 ≤ j ≤ k} and put u_{j} = θ_{j}u, for every j = 0, …, k. We have u = ∑_{j=0}^{k} u_{j} and supp u_{j} ⊂ G̃_{j}, for every j = 0, …, k.

So, it suffices to show that any u_{j} belongs in W_{0}^{1}L_{M}(Ω ∩ G_{j}). Since \overline{G̃_{0}} ⊂ Ω, as in the case K ⊂ Ω above we prove that u_{0} ∈ W_{0}^{1}L_{M}(Ω ∩ G_{j}).

Since u_{j} ∈ W^{1}L_{M}(Ω ∩ G_{j}) for all j ≥ 1 we claim that ũ_{j} ∈ W^{1}L_{M}(ℝ^{N}) (indeed: the function ũ_{j} lies in W_{0}^{1,1}(ℝ^{N}) due to u_{j} ∈ W_{0}^{1,1}(Ω ∩ G_{j}). Moreover, ∇ũ_{j} = ˜∇u_{j} in the distributional sense and a.e. on ℝ^{N}. On the other hand, ũ_{j} ∈ L_{M}(ℝ^{N}) and ∇ũ_{j} ∈ (L_{M}(ℝ^{N}))^{N}, because u_{j} ∈ W^{1}L_{M}(Ω∩G_{j})). Let K_{j} = supp u_{j} and u_{j,t} = ũ_{j}(x − ty_{j}), with 0 < t < min{1, |y_{j}|^{−1} dist(G̃_{j}, G_{j}^{c})}, where y_{j} be the nonzero vector associated to the set G_{j} (see Definition 2.1). We claim that supp u_{j,t} ⊂ Ω ∩ G_{j}, for all t satisfying 0 < t < min{1, |y_{j}|^{−1} dist(G̃_{j}, G_{j}^{c})}. In fact, by the segment property, we have

supp u_{j,t} = K_{j} + ty_{j} ⊂ (G_{j} ∩ \overline{Ω}) + ty_{j} ⊂ Ω.

On the other hand, let x ∈ supp u_{j,t}. Then dist(x, G̃_{j}) ≤ dist(x, x − ty_{j}) + dist(x − ty_{j}, K_{j}) + dist(K_{j}, G̃_{j}) = dist(x, x − ty_{j}), which implies that,

dist(x, G̃_{j}) ≤ dist(x, x − ty_{j}) = |ty_{j}|.

Then, dist(x, G̃_{j}) < dist(G̃_{j}, G_{j}^{c}), hence x ∈ G_{j}. Since u_{j,t} ∈ W_{0}^{1}L_{M}(ℝ^{N}) and supp u_{j,t} ⊂ Ω ∩ G_{j}, in virtue of lemma 1.5 of [12], we see that u_{j,t} → 0 in W^{1}L_{M}(Ω ∩ G_{j}) for σ(ΠL_{M}, ΠL\overline{M}) as t → 0. Moreover, by using lemma 1.6 of [12] we can approximate u_{j,t} by a sequences of elements of D(Ω ∩ G_{j})), W^{1}L_{M}(Ω ∩ G_{j}) for σ(ΠL_{M}, ΠL\overline{M}), hence gives the result.

Step 2 We shall prove that, W_{0}^{1}L_{M}(Ω) ⊂ W_{0}^{1,1}(Ω) ∩ W^{1}L_{M}(Ω). Let u ∈ W_{0}^{1}L_{M}(Ω), by theorem 1.4 of [13] there exists u_{n} ∈ D(Ω) and λ > 0 such that

∫_{Ω} M(\frac{D^{α}u_{n} − D^{α}u}{λ}) dx → 0 as n → ∞ ∀ |α| ≤ 1.

By using Jensen’s inequality, we have

M(\frac{1}{meas(Ω)} ∫_{Ω}(\frac{D^{α}u_{n} − D^{α}u}{λ}) dx) ≤ \frac{1}{meas(Ω)} ∫_{Ω} M(\frac{D^{α}u_{n} − D^{α}u}{λ}) dx

for n large enough. Then,

M(\frac{1}{meas(Ω)} ∫_{Ω}(\frac{D^{α}u_{n} − D^{α}u}{λ}) dx) → 0, as n → ∞ ∀ |α| ≤ 1,

which gives, since M^{−1} is right continuous in ℝ^{+},

∫_{Ω} |D^{α}u_{n} − D^{α}u| dx → 0 as n → ∞ ∀ |α| ≤ 1.

This completes the proof. □

Let M and P be two N-functions such that P << M. Consider the Leray Lions operator A defined from D(A) ⊂ W_{0}^{1}L_{M}(Ω) → W^{−1}L\overline{M}(Ω) by

Au = − div(a(x, u, ∇u)),

where a : Ω×ℝ×ℝ^{n} → ℝ^{n} is a Carathéodory function satisfying for a.e. x ∈ Ω, all s ∈ ℝ and all ξ ≠ ξ^{∗} ∈ ℝ^{n}:

(H1’) |a(x, s, ξ)| ≤ c(x) + k_{1}\overline{P}^{−1}M(k_{2}|s|) + k_{3}\overline{M}^{−1}M(k_{4}|ξ|)

(H2’) a(x, s, ξ)ξ ≥ αM(\frac{|ξ|}{β}) for some positive constants k_{1}, …, k_{4}, α and β, where c(x) belongs to E\overline{M}(Ω).

Consider the nonlinear boundary-value problem

− div(a(x, u, ∇u)) = f in Ω

u ≡ θ in ∂Ω. (3.1)

Our objective is the following.

Theorem 3.2 Let u ∈ W^{1}L_{M}(Ω) denotes a weak solution of (3.1) and assume that f is in L^{∞}(Ω). Then sup_{∂Ω} θ < ∞ implies that u is bounded from above i.e.

sup_{Ω} u ≤ const(sup_{∂Ω} θ, ∥u∥_{L^{1}(Ω)}, n, |Ω|, M, ∥f∥_{L^{∞}(Ω)}, α, β) < ∞.

Remark 3.1 Note that the statement of Theorem 3.2 holds for any N-function M. In particular for the following critical cases:

M(t) = t log(1 + t) and M(t) = e^{t} − t − 1.

We state the following lemmas which are needed below.

Lemma 3.1 ([5]) Let k_{0} > 0, γ > 0, ε > 0 and α ∈ [0, 1 + ε] denote constants and suppose that u ∈ L^{1}(Ω) satisfies the estimate

∫_{A_{k}}(u − k)dx ≤ γk^{α}|A_{k}|^{1+ε},

for all k ≥ k_{0}, where A_{k} denotes the set of points x ∈ Ω for which u(x) > k. Then sup_{Ω} u is bounded by a finite constant depending on γ, ε, α, k_{0} and ∥u∥_{L^{1}(A_{k_{0}} )}.

Proof of Theorem 3.2 Let k_{0} = sup_{∂Ω} θ < ∞ and let u be a weak solution of (3.1). Let us remark that by the lemma 3.1, it suffices to show that

∫_{A_{k}}(u − k)dx ≤ const(n, |Ω|, M, ∥f∥_{L^{∞}(Ω)}, α, β)|A_{k}|^{1+ \frac{1}{n}} ∀k ≥ k_{0}. (3.2)

Observe that

\begin{matrix}∫_{A_{k}} f(u − k)dx ≤ & |A_{k}|^{1/n}( ∫_{A_{k}} |u − k|^{\frac{n}{n−1}} dx)^{\frac{n−1}{n}} \\ ≤ & c(n)|A_{k}|^{1/n} ∫_{A_{k}} |∇u|dx. \end{matrix}  (3.3)

On the other hand, we have

\begin{matrix}∫_{A_{k}} |∇u|dx ≤ & ∫_{A_{k}} β \frac{|∇u|}{β}dx ≤ ∫_{A_{k}}(\overline{M}(β) + M(\frac{|∇u|}{β}))dx \\ ≤ & \overline{M}(β)|A_{k}| + ∫_{A_{k}} M(\frac{|∇u|}{β})dx. \end{matrix}  (3.4)

Moreover by theorem 3.1, the function φ = max(u − k, 0) lies in the space W_{0}^{1}L_{M}(Ω). Hence φ is admissible in (3.1) and we obtain,

∫_{A_{k}} M(\frac{|∇u|}{β})dx ≤ \frac{1}{α} ∫_{A_{k}} f(u − k)dx.  (3.5)

The right hand side of this inequality can be estimated with Hölder’s inequality and the imbedding W_{0}^{1.1}(Ω) ↪ L^{1^{∗}}(Ω) as follows:

\begin{matrix}\frac{1}{α} ∫_{A_{k}} f(u − k)dx ≤ & \frac{1}{α}∥f∥_{∞}|A_{k}|^{1/n}(∫_{A_{k}} |u − k|^{\frac{n}{n−1}} dx)^{\frac{n−1}{n}} \\ ≤ & \frac{2β}{α} c(n)∥f∥_{∞}|A_{k}|^{1/n} ∫_{A_{k}} \frac{|∇u|}{2β}dx. \end{matrix}

Using Young’s inequality we can write

\begin{matrix}\frac{1}{α} ∫_{A_{k}} f(u − k)dx ≤ & \overline{M}(\frac{2β}{α} c(n)∥f∥_{∞}|A_{k}|^{1/n})|A_{k}| + ∫_{A_{k}} M(\frac{|∇u|}{2β})dx \\ ≤ & \overline{M}(\frac{2β}{α} c(n)∥f∥_{∞}|A_{k}|^{1/n})|A_{k}| + \frac{1}{2} ∫_{A_{k}} M(\frac{|∇u|}{β})dx. \end{matrix} (3.6)

Then the conclusion follow immediately from (3.3)–(3.6).

Remark 3.2 The method used in the proof of the above theorem gives also inf_{Ω} u > −∞ provided that θ is bounded from below. In particular, boundedness of θ implies u ∈ L^{∞}(Ω) (compare with remark 1.1 [6]).

Example Let M(t) = e^{|t|} − |t| − 1, we set a(x, s, ξ) = (a_{i}(x, s, ξ))_{1≤i≤n} such that

a_{i}(x, s, ξ) = \begin{cases}\frac{e^{|ξ_{i}|}−|ξ_{i}|−1}{|ξ_{i}|} sign ξ_{i} & if |ξ_{i}| ≠ 0 \\ 0 & if |ξ_{i}| = 0 . \end{cases}

Then M(t) and a(x, s, ξ) satisfy the conditions (1.3) and (1.4). Note that the N-function M(t) does not satisfy the Δ_{2}-condition.

Remark 3.3 Let m(t) the right derivative of M(t). Then m(t) and a(x, s, ξ) don’t satisfy the condition (H_{2}). Indeed, take ξ = (0, …, 0, n, 0, …, 0), n ∈ ℕ. Then we have a(x, s, ξ)ξ = e^{n} − n − 1, and |ξ|m(|ξ|) = n(e^{n} − 1). But for all constant C > 0 there exists n large enough, such that \frac{e^{n}−n−1}{n(e^{n}−1)} < C.

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Lahsen Aharouch (e-amil: lahrouche@caramail.com) Elhoussine Azroul (e-mail: elazroul@caramail.com) Abdelmoujib Benkirane (e-mail: abenkirane@fsdmfes.ac.ma) Département de Mathématiques et Informatique Faculté des Sciences Dhar-Mahraz B.P. 1796 Atlas Fès, Maroc