We show that this problem has an unique solution which is the solution to the variational inequality arising in the elasto-plastic torsion problem, associated with and operatorA

Electronic Journal of Differential Equations, Vol. 2006(2006), No. 156, pp. 1–7.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp)

ELASTO-PLASTIC TORSION PROBLEM AS AN INFINITY LAPLACE’S EQUATION

AHMED ADDOU, ABDELUAAB LIDOUH, BELKASSEM SEDDOUG

Abstract. In this paper, we study a perturbed infinity Laplace’s equation, the perturbation corresponds to an Leray-Lions operator with no coercivity assumption. We consider the case where data are distributions orL¹elements.

We show that this problem has an unique solution which is the solution to the variational inequality arising in the elasto-plastic torsion problem, associated with and operatorA.

1. Introduction

Given a bounded open subset Ω ofR^N,N ≥1, we consider the Dirichlet Problem Au−∆_∞u=f in Ω,

u= 0 on∂Ω, (1.1)

where ∆_∞u=u_xiu_xju_xixj (see [3]),f inL¹(Ω) orW^−1,p0(Ω) andAis a Leray-Lions operator with no coercivity assumption, i.e.

Av=−div(a(x,∇v(x)))

where a : Ω×R^N → R^N is a Caratheodory function satisfying the following as- sumptions:

For almost everyx∈Ω and for all ξ, η∈R^N, (ξ6=η), one has:

a(x, ξ)ξ≥0, (1.2)

|a(x, ξ)| ≤β

h(x) +|ξ|^p−1

, (1.3)

a(x, ξ)−a(x, η)

(ξ−η)>0 (1.4)

with 1< p <+∞,β >0,h∈L^p0(Ω) (p⁰ denotes the conjugate exponent ofp, i.e:

1

p+_p¹0 = 1).

By a solution to 1.1 we will mean a variational solution in the sense which extends that given in ([3]) and ([9]), that is, a functionuwhich is the limit of the sequence (un) of solutions to the Dirichlet problems

Au_n−∆_nu_n=f in Ω, un= 0 on∂Ω,

2000Mathematics Subject Classification. 35J70, 35J85, 74C05.

Key words and phrases. Infinity Laplace equation; elasto-plastic torsion problem;

variational inequality.

c

2006 Texas State University - San Marcos.

Submitted October 11, 2006. Published December 18, 2006.

1

asn→ ∞, where ∆n is then-Laplacian operator (∆nv= div(|∇v|ⁿ⁻²∇v).

We show that in the variational case (f ∈W^−1,p0(Ω)), the sequence (un) con- verges to the unique solution to the variational inequality

hAu, v−ui ≥ hf, v−ui, for allv∈ K, u∈ K.

WhereKis the bounded convex cone ofW₀^1,p(Ω) defined as:

K={v∈W₀^1,p(Ω) :|∇v(x)| ≤1 a.e. in Ω},

and in the case f ∈L¹(Ω), the sequence (u_n) converges to the unique solution to the problem

hAu, Tk(v−u)i ≥ Z

Ω

f T_k(v−u)dx, for allv∈ K, u∈ K, for allk >0.

WhereTk:R→Ris the cut function defined as T_k(s) =

(s if|s| ≤k ksign(s) if|s|> k.

hereh., .idenotes the duality pairing betweenW^−1,p0(Ω) andW₀^1,p(Ω).

Our approach is also inscribed among the techniques of “the increase of power”, first introduced by Boccardo and Murat in [4], where they approached the problem

hAu, v−ui ≥ hf, v−ui, for allv∈ K0, u∈ K0={v∈W₀^1,p(Ω) :|v(x)| ≤1 a.e. in Ω}, by the sequence of the Dirichlet equations

Au_n− |un|ⁿ⁻¹u_n=f inD⁰(Ω), un∈W₀^1,p(Ω)∩Lⁿ(Ω), wheref ∈W^−1,p0(Ω) and Ais modelled on thep-Laplacian.

Then in [5], Dall’Aglio and Orsina generalized this result by considering in- creasing powers depending of a certain Caratheodory function satisfying the sign condition and an integrability assumption.

Then finally in [2] the authors extended this result to the case where increas- ing powers are multiplied by a quantity depending on the gradient and verifying adequate conditions, they examine the two cases,f inL¹(Ω) and inW^−1,p0(Ω).

In this paper we examine the case where the increasing powers carry on the gradients and not on quantities independent of the gradient.

2. The variational case

Letf ∈W^−1,p0(Ω), 1< p <+∞. For all integern≥p, we consider the Dirichlet problem

Au_n−∆_nu_n=f in Ω,

u_n∈W₀^1,n(Ω). (2.1)

It is known [7, 8] that, under assumptions (1.2)–(1.4), the problem (2.1) has an unique solutionun, in the following sense:

∀v∈W₀^1,n(Ω) : Z

Ω

[a(x,∇un)∇v+|∇un|ⁿ⁻²∇un∇v]dx=hf, vi. (2.2) In the sequelW₀^1,p(Ω) is equipped with its usual norm

kvk_W1,p

0 (Ω)=hZ

Ω

|∇v|^pdxi^1/p Let us now, state our first main result.

Theorem 2.1. Let f ∈W^−1,p0(Ω),1< p <+∞. Under assumptions (1.2)–(1.4), ifun designates the solution to the problem (2.1), then the sequence(un)converges strongly inW₀^1,p(Ω), to the unique solution uto the problem

hAu, v−ui ≥ hf, v−ui, for all v∈ K,

u∈ K. (2.3)

Proof of Theorem 2.1.

A priori estimate. Withu_n as a test function in (2.2), we get Z

Ω

a(x,∇un)∇undx+ Z

Ω

|∇un|ⁿdx=hf, uni ≤ kfk−1,p⁰kunk1,p

hence

Z

Ω

|∇un|ⁿdx≤ckunk1,p for alln≥p . (2.4) In the sequelc, c₁, c₂. . .. designate arbitrary constants.

From (2.4), and by splittingR

Ω|∇un|^pdxas Z

Ω

|∇un|^pdx= Z

[|∇un|≤1]

|∇un|^pdx+ Z

[|∇un|>1]

|∇un|^pdx,

one deduces that Z

Ω

|∇un|^pdx≤ |Ω|+c[

Z

Ω

|∇un|^pdx]^1p for alln≥p and so

Z

Ω

|∇un|^pdx≤c for alln≥p . (2.5) Thereafter,

Z

Ω

|∇un|ⁿdx≤c ∀n and Z

Ω

|∇un|^qdx≤c ∀q, ∀n≥q. (2.6) Therefore, one can construct a subsequence, still denoted by (un)n, such that

u_n* u weakly inW₀^1,q(Ω) and uniformly in ¯Ω, (2.7) for someu∈W₀^1,q(Ω)∩L^∞(Ω), for allq >1. More precisely, we have

u∈W₀^1,∞(Ω) and k∇uk_∞≤1. (2.8)

Indeed, from (2.6) and (2.7), one has k∇uk_∞= lim

q→∞k∇ukq ≤ lim

q→∞ lim inf

n→∞ k∇unkq

≤ lim

q→∞c^1q = 1.

Almost everywhere convergence of gradients. With v = un −u, as a test function in (2.2), and using the fact that

∇un(∇un− ∇u)≥0 in the set{|∇un| ≥ |∇u|}, one has

hAun, un−ui+ Z

{|∇un|<|∇u|}

|∇un|ⁿ⁻²∇un(∇un− ∇u)dx≤εn, (2.9) We will denote byεn any quantity which converges to zero asntends to infinity.

Letε >0, for the second term on the left in (2.9), one puts

A1={|∇un|<|∇u|and|∇un| ≤1−ε}, A2={1−ε <|∇un|<|∇u|}

and so we have Z

A1

|∇un|ⁿ⁻²∇un(∇un− ∇u)dx=σn,ε, (2.10) whereσn,εdenotes a quantity depending onnandε, such that, for any fixedε >0, σn,ε→0, asn→ ∞, and which may change from line to line. Also

Z

A₂

|∇u_n|ⁿ⁻²∇u_n(∇u_n− ∇u)dx

= Z

A₂

|∇un|ⁿ⁻²(|∇un|²− |∇u|²)dx+ Z

A₂

|∇un|ⁿ⁻²∇u(∇u− ∇un)dx

=qn+In,

(2.11)

where the quantityInis nonnegative, andqn∈[−2ε|Ω|,0]. Combining (2.9), (2.10) and (2.11), one gets

hAu_n, u_n−ui ≤σ_n,ε+ 2ε|Ω|,∀ε >0 On the other hand,hAu, u_n−ui →0, asn→ ∞, so that

0≤ hAun−Au, u_n−ui ≤σ_n,ε+ 2ε|Ω|,∀ε >0.

Passing to the limit asn→ ∞, for any fixedε, one has 0≤lim inf

n→∞hAun−Au, un−ui ≤lim sup

n→∞

hAun−Au, un−ui ≤2ε|Ω| ∀ε >0.

By the arbitrariness of ε(and since hAun−Au, un−ui does not depend onε) it follows that

hAun−Au, un−ui →0 asn→ ∞. (2.12) Which implies, thanks to (1.4), that (for a subsequence),

(a(x,∇un)−a(x,∇u))(∇un− ∇u)→0 a.e. in Ω.

For a fixedk >1, we put X= \

q∈N

[

n≥q

{|∇un| ≥k}, and its complementY = [

q∈N

\

n≥q

{|∇un|< k},

for allx∈Y, the sequence (∇un(x)) is bounded inR^N, so

∇u_n(x)→ξ

for a subsequence and someξ∈R^N, while (1.4) and the continuity ofa(x, .), implies thatξ=∇u(x), we can then conclude that

∇un(x)→ ∇u(x) for allx∈Y.

To show the almost everywhere convergence of (∇un), it suffices to prove that meas(X) = 0. In deed, from (2.6), one has

meas{|∇un| ≥k}= Z

{|∇un|≥k}

1dx≤ c

kⁿ. (2.13)

SinceX ⊂S

n≥q{|∇un| ≥k}, for allq, one deduces that meas(X)≤X

n≥q

meas{|∇un| ≥k} →0 asq→ ∞.

Strong convergence in W₀^1,p(Ω). Thanks to Vitali’s theorem, it suffices to show the equi-integrability of (|∇un|^p) inL¹(Ω), what follows from (2.6) withq=p+ 1.

Indeed for a measurable subsetE of Ω, one has Z

E

|∇un|^pdx≤Z

E

|∇un|^p+1dx_p+1^p Z

E

1dx_p+1¹

≤c meas(E)_p+1¹ .

The function uis solution to problem(2.3). Letv∈ Kand 0< θ <1, taking z=u_n−θT_k(v) as a test function in (2.2), one gets

hAu_n, zi+ Z

Ω

|∇u_n|ⁿ⁻²∇u_n∇zdx=hf, zi While noticing that

Z

{|∇un|≥θ|∇T_k(v)|}

|∇u_n|ⁿ⁻²∇u_n(∇u_n−θ∇T_k(v))dx≥0 one has

hAun, zi+ Z

{|∇un|<θ|∇Tk(v)|}

|∇un|ⁿ⁻²∇un∇zdx≤ hf, zi

Passing to the limit as n → ∞, and using standard result about Caratheodory functions satisfying (1.3), one gets

hAu, u−θTk(v)i ≤ hf, u−θTk(v)i

The result is then obtained while passing to the limit asθ→1 andk→ ∞.

3. The casef ∈L¹(Ω)

In this section, we suppose thatf ∈L¹(Ω), as in the previous section. Now we prove our second main result.

Theorem 3.1. Let f ∈L¹(Ω), 1 < p <+∞. Under assumptions (1.2)–(1.4), if un (n > N) designates the solution to the problem (2.1), then the sequence (un) converges strongly inW₀^1,p(Ω), to the unique solutionuto the problem

hAu, Tk(v−u)i ≥ Z

Ω

f Tk(v−u)dx for allv∈ K, u∈ K, for allk >0.

(3.1) Proof of Theorem 3.1. According to the previous section, it is clear that the estimate (2.6) permits to show that the sequence (un) converges in W₀^1,p(Ω) and uniformly in ¯Ω (for a subsequence) tousatisfying (2.8).

We are going to prove (2.6) and the fact thatuis the solution to (3.1).

A priori estimate. Withun (n > N) as a test function in (2.2), we get Z

Ω

a(x,∇un)∇undx+ Z

Ω

|∇un|ⁿdx= Z

Ω

f u_ndx≤ kfk1kunk∞

Letq > N (fixed), by splittingR

Ω|∇un|^qdxas Z

Ω

|∇un|^qdx= Z

{|∇u_n|<1}

|∇un|^qdx+ Z

{|∇u_n|≥1}

|∇un|^qdx

and using Sobolev’s inequality [1], one has Z

Ω

|∇un|^qdx≤c ∀n≥q; (3.2)

therefore,

Z

Ω

|∇un|ⁿdx≤c ∀n > N .

It follows that the estimate (3.2) holds for all q > 1, what leads to the estimate (2.6).

The function uis solution to problem(3.1). Letv∈ Kand 0< θ <1, taking z=Tk(un−θv) as a test function in (2.2), one gets

hAun, zi+ Z

Ω

|∇un|ⁿ⁻²∇un∇zdx= Z

Ω

f zdx

While noticing that Z

{|∇un|≥θ|∇v|}

|∇un|ⁿ⁻²∇un∇Tk(un−θv)dx≥0 one has

hAun, zi+ Z

{|∇un|<θ|∇v|}

|∇un|ⁿ⁻²∇un∇zdx≤ Z

Ω

f z dx

Passing to the limit asn→ ∞, one gets hAu, T_k(u−θv)i ≤

Z

Ω

f T_k(u−θv)dx The result is obtained when passing to the limit asθ→1.

Remark 3.2. Sinceu∈W₀^1,∞(Ω), the problem can be formulated in this space by choosingK={v∈W₀^1,∞(Ω) :k∇v(x)k∞≤1}, what permits to write the problem (3.1) without truncation operator, and simplify the proof of the stepThe function u is solution to the problem (3.1). But traditionally (see for example [6]), the elasto- plastic torsion problem is written withK={v∈W₀^1,p(Ω) :|∇v(x)| ≤1 a.e. in Ω}, it’s why we have done this choice.

Acknowledgement. The authors would like to thank the anonymous referee for his/her interesting remarks.

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Ahmed Addou

Universit´e Mohammed premier, Facult´e des sciences, Oujda, Maroc E-mail address:[email protected]

Abdeluaab Lidouh

Universit´e Mohammed premier, Facult´e des sciences, Oujda, Maroc E-mail address:[email protected]

Belkassem Seddoug

Universit´e Mohammed premier, Facult´e des sciences, Oujda, Maroc E-mail address:[email protected]