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 or L1 elements.
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 operator A.
1. Introduction
Given a bounded open subset Ω of RN , N ≥ 1, we consider the Dirichlet Problem
Au − ∆∞ u = f
u=0
in Ω,
(1.1)
on ∂Ω,
0
where ∆∞ u = uxi uxj uxi xj (see [3]), f in L1 (Ω) or W −1,p (Ω) and A is a Leray-Lions
operator with no coercivity assumption, i.e.
Av = − div(a(x, ∇v(x)))
N
N
where a : Ω × R → R is a Caratheodory function satisfying the following assumptions:
For almost every x ∈ Ω and for all ξ, η ∈ RN , (ξ 6= η), one has:
a(x, ξ)ξ ≥ 0,
|a(x, ξ)| ≤ β h(x) + |ξ|p−1 ,
a(x, ξ) − a(x, η) (ξ − η) > 0
p0
0
(1.2)
(1.3)
(1.4)
with 1 < p < +∞, β > 0, h ∈ L (Ω) (p denotes the conjugate exponent of p, i.e:
1
1
p + p0 = 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 function u which is the limit of the sequence
(un ) of solutions to the Dirichlet problems
Aun − ∆n un = f in Ω,
un = 0
on ∂Ω,
2000 Mathematics 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
2
A. ADDOU, A. LIDOUH, B. SEDDOUG
EJDE-2006/156
as n → ∞, where ∆n is the n-Laplacian operator (∆n v = div(|∇v|n−2 ∇v).
0
We show that in the variational case (f ∈ W −1,p (Ω)), the sequence (un ) converges to the unique solution to the variational inequality
hAu, v − ui ≥ hf, v − ui, for all v ∈ K,
u ∈ K.
Where K is the bounded convex cone of W01,p (Ω) defined as:
K = {v ∈ W01,p (Ω) : |∇v(x)| ≤ 1 a.e. in Ω},
and in the case f ∈ L1 (Ω), the sequence (un ) converges to the unique solution to
the problem
Z
hAu, Tk (v − u)i ≥
f Tk (v − u)dx, for all v ∈ K,
Ω
u ∈ K,
for all k > 0.
Where Tk : R → R is the cut function defined as
(
s
if |s| ≤ k
Tk (s) =
k sign(s) if |s| > k.
0
here h., .i denotes the duality pairing between W −1,p (Ω) and W01,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,
u ∈ K0 = {v ∈
W01,p (Ω)
for all v ∈ K0 ,
: |v(x)| ≤ 1 a.e. in Ω},
by the sequence of the Dirichlet equations
Aun − |un |n−1 un = f
in D0 (Ω),
un ∈ W01,p (Ω) ∩ Ln (Ω),
0
where f ∈ W −1,p (Ω) and A is modelled on the p-Laplacian.
Then in [5], Dall’Aglio and Orsina generalized this result by considering increasing 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 increasing powers are multiplied by a quantity depending on the gradient and verifying
0
adequate conditions, they examine the two cases, f in L1 (Ω) and in W −1,p (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
0
Let f ∈ W −1,p (Ω), 1 < p < +∞. For all integer n ≥ p, we consider the Dirichlet
problem
Aun − ∆n un = f in Ω,
(2.1)
un ∈ W01,n (Ω).
EJDE-2006/156
ELASTO-PLASTIC TORSION PROBLEM
3
It is known [7, 8] that, under assumptions (1.2)–(1.4), the problem (2.1) has an
unique solution un , in the following sense:
Z
∀v ∈ W01,n (Ω) : [a(x, ∇un )∇v + |∇un |n−2 ∇un ∇v]dx = hf, vi.
(2.2)
Ω
In the sequel W01,p (Ω) is equipped with its usual norm
hZ
i1/p
kvkW 1,p (Ω) =
|∇v|p dx
0
Ω
Let us now, state our first main result.
0
Theorem 2.1. Let f ∈ W −1,p (Ω), 1 < p < +∞. Under assumptions (1.2)–(1.4),
if un designates the solution to the problem (2.1), then the sequence (un ) converges
strongly in W01,p (Ω), to the unique solution u to the problem
hAu, v − ui ≥ hf, v − ui,
u ∈ K.
for all v ∈ K,
(2.3)
Proof of Theorem 2.1.
A priori estimate. With un as a test function in (2.2), we get
Z
Z
a(x, ∇un )∇un dx +
|∇un |n dx = hf, un i ≤ kf k−1,p0 kun k1,p
Ω
Ω
hence
Z
|∇un |n dx ≤ ckun k1,p
for all n ≥ p .
(2.4)
Ω
In the sequel c, c1 , c2 . . . . designate
R arbitrary constants.
From (2.4), and by splitting Ω |∇un |p dx as
Z
Z
Z
|∇un |p dx =
|∇un |p dx +
[|∇un |≤1]
Ω
one deduces that
Z
[|∇un |>1]
Z
1
|∇un |p dx ≤ |Ω| + c[ |∇un |p dx] p
Ω
|∇un |p dx,
for all n ≥ p
Ω
and so
Z
|∇un |p dx ≤ c for all n ≥ p .
(2.5)
Ω
Thereafter,
Z
|∇un |n dx ≤ c ∀n
Z
and
Ω
|∇un |q dx ≤ c ∀q, ∀n ≥ q.
(2.6)
Ω
Therefore, one can construct a subsequence, still denoted by (un )n , such that
weakly in W01,q (Ω) and uniformly in Ω̄,
un * u
for some u ∈
W01,q (Ω)
(2.7)
∩ L∞ (Ω), for all q > 1. More precisely, we have
u ∈ W01,∞ (Ω)
and k∇uk∞ ≤ 1.
(2.8)
Indeed, from (2.6) and (2.7), one has
k∇uk∞ = lim k∇ukq ≤ lim
q→∞
q→∞
1
lim inf k∇un kq ≤ lim c q = 1.
n→∞
q→∞
4
A. ADDOU, A. LIDOUH, B. SEDDOUG
EJDE-2006/156
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
Z
hAun , un − ui +
|∇un |n−2 ∇un (∇un − ∇u)dx ≤ εn ,
(2.9)
{|∇un |<|∇u|}
We will denote by εn any quantity which converges to zero as n tends to infinity.
Let ε > 0, for the second term on the left in (2.9), one puts
A1 = {|∇un | < |∇u| and |∇un | ≤ 1 − ε},
and so we have
Z
A2 = {1 − ε < |∇un | < |∇u|}
|∇un |n−2 ∇un (∇un − ∇u)dx = σn,ε ,
(2.10)
A1
where σn,ε denotes a quantity depending on n and ε, such that, for any fixed ε > 0,
σn,ε → 0, as n → ∞, and which may change from line to line. Also
Z
|∇un |n−2 ∇un (∇un − ∇u)dx
A2
Z
Z
(2.11)
n−2
2
2
=
|∇un |
(|∇un | − |∇u| )dx +
|∇un |n−2 ∇u(∇u − ∇un )dx
A2
A2
= qn + In ,
where the quantity In is nonnegative, and qn ∈ [−2ε|Ω|, 0]. Combining (2.9), (2.10)
and (2.11), one gets
hAun , un − ui ≤ σn,ε + 2ε|Ω|, ∀ε > 0
On the other hand, hAu, un − ui → 0, as n → ∞, so that
0 ≤ hAun − Au, un − ui ≤ σn,ε + 2ε|Ω|, ∀ε > 0.
Passing to the limit as n → ∞, for any fixed ε, one has
0 ≤ lim inf hAun − Au, un − ui ≤ lim suphAun − Au, un − ui ≤ 2ε|Ω|
n→∞
∀ε > 0.
n→∞
By the arbitrariness of ε (and since hAun − Au, un − ui does not depend on ε) it
follows that
hAun − Au, un − ui → 0 as n → ∞.
(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 fixed k > 1, we put
\ [
[ \
X=
{|∇un | ≥ k}, and its complement Y =
{|∇un | < k},
q∈N n≥q
q∈N n≥q
for all x ∈ Y , the sequence (∇un (x)) is bounded in RN , so
∇un (x) → ξ
N
for a subsequence and some ξ ∈ R , while (1.4) and the continuity of a(x, .), implies
that ξ = ∇u(x), we can then conclude that
∇un (x) → ∇u(x)
for all x ∈ Y.
EJDE-2006/156
ELASTO-PLASTIC TORSION PROBLEM
5
To show the almost everywhere convergence of (∇un ), it suffices to prove that
meas(X) = 0. In deed, from (2.6), one has
Z
c
meas{|∇un | ≥ k} =
(2.13)
1dx ≤ n .
k
{|∇un |≥k}
S
Since X ⊂ n≥q {|∇un | ≥ k}, for all q, one deduces that
X
meas{|∇un | ≥ k} → 0 as q → ∞.
meas(X) ≤
n≥q
Strong convergence in W01,p (Ω). Thanks to Vitali’s theorem, it suffices to show
the equi-integrability of (|∇un |p ) in L1 (Ω), what follows from (2.6) with q = p + 1.
Indeed for a measurable subset E of Ω, one has
Z
p Z
1
Z
p+1
p+1
1
|∇un |p dx ≤
|∇un |p+1 dx
1 dx
≤ c meas(E) p+1 .
E
E
E
The function u is solution to problem (2.3). Let v ∈ K and 0 < θ < 1, taking
z = un − θTk (v) as a test function in (2.2), one gets
Z
hAun , zi +
|∇un |n−2 ∇un ∇zdx = hf, zi
Ω
While noticing that
Z
|∇un |n−2 ∇un (∇un − θ∇Tk (v))dx ≥ 0
{|∇un |≥θ|∇Tk (v)|}
one has
Z
|∇un |n−2 ∇un ∇zdx ≤ hf, zi
hAun , zi +
{|∇un |<θ|∇Tk (v)|}
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 as θ → 1 and k → ∞.
3. The case f ∈ L1 (Ω)
In this section, we suppose that f ∈ 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 in W01,p (Ω), to the unique solution u to the problem
Z
hAu, Tk (v − u)i ≥
f Tk (v − u)dx for all v ∈ K,
(3.1)
Ω
u ∈ K, for all k > 0.
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 W01,p (Ω) and
uniformly in Ω̄ (for a subsequence) to u satisfying (2.8).
We are going to prove (2.6) and the fact that u is the solution to (3.1).
6
A. ADDOU, A. LIDOUH, B. SEDDOUG
EJDE-2006/156
A priori estimate. With un (n > N ) as a test function in (2.2), we get
Z
Z
Z
a(x, ∇un )∇un dx +
|∇un |n dx =
f un dx ≤ kf k1 kun k∞
Ω
Ω
Let q > N (fixed), by splitting
Z
Z
|∇un |q dx =
Ω
q
R
Ω
|∇un | dx as
|∇un |q dx +
{|∇un |<1}
Ω
Z
|∇un |q dx
{|∇un |≥1}
and using Sobolev’s inequality [1], one has
Z
|∇un |q dx ≤ c ∀n ≥ q;
(3.2)
Ω
therefore,
Z
|∇un |n dx ≤ c ∀n > N .
Ω
It follows that the estimate (3.2) holds for all q > 1, what leads to the estimate
(2.6).
The function u is solution to problem (3.1). Let v ∈ K and 0 < θ < 1, taking
z = Tk (un − θv) as a test function in (2.2), one gets
Z
Z
n−2
hAun , zi +
|∇un |
∇un ∇zdx =
f zdx
Ω
While noticing that
Z
Ω
|∇un |n−2 ∇un ∇Tk (un − θv)dx ≥ 0
{|∇un |≥θ|∇v|}
one has
Z
hAun , zi +
|∇un |n−2 ∇un ∇zdx ≤
{|∇un |<θ|∇v|}
Z
f z dx
Ω
Passing to the limit as n → ∞, one gets
Z
hAu, Tk (u − θv)i ≤
f Tk (u − θv) dx
Ω
The result is obtained when passing to the limit as θ → 1.
Remark 3.2. Since u ∈ W01,∞ (Ω), the problem can be formulated in this space by
choosing K = {v ∈ W01,∞ (Ω) : k∇v(x)k∞ ≤ 1}, what permits to write the problem
(3.1) without truncation operator, and simplify the proof of the step The function u
is solution to the problem (3.1). But traditionally (see for example [6]), the elastoplastic torsion problem is written with K = {v ∈ W01,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.
EJDE-2006/156
ELASTO-PLASTIC TORSION PROBLEM
7
References
[1] R. Adams. Sobolev spaces. Academic Press, New York, 1975.
[2] A. Benkirane, A. EL Mahi, D. Meskine. On the limit of some penalized problems involving
increasing powers. Asymptotic Analysis, Volume 36, Numbers 3–4 (2003), pp. 303–317.
[3] T. Bhattacharya, E. DiBenedetto, J. Manfredi. Limits as p → ∞ of ∆p (u) = f and related
extremal problems. Rend. Sem. Mat. Univ. Pol. Torino, Fascicolo Speciale Nonlinear PDE’s,
(1989), 15–68.
[4] L. Boccardo, F. Murat. Increase of power leads to bilateral problems. Composite Media and
Homogenization Theory, G. Dal Maso and G. F. Dell’ Antonio. eds., World Scientific, Singapore, 1995, pp. 113–123.
[5] A. Dall’Aglio, L. Orsina. On the limit of some nonlinear elliptic equations involving increasing
powers. Asympt. Anal., 14 (1997), 49–71.
[6] D. Kinderlehrer, G. Stampacchia. An introduction to variational inequalities and their applications. Academic Press, New York, 1980.
[7] L. Leray, J. L. Lions. Quelques résultats de Visik sur les problèmes non linéaires par les
méthodes de Minty-Browder. Bull. Soc. Math. France, 93 (1965), 97–107.
[8] J. L. Lions. Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod
et Gauthiers-Villars (1969).
[9] E. Rosset. Symmetry and convexity of level sets of solutions to the infinity Laplace’s equation.
Electron. J. Differential Equations, vol. 1998, (1998) No. 34, pp. 1–12.
Ahmed Addou
Université Mohammed premier, Faculté des sciences, Oujda, Maroc
E-mail address: addou@sciences.univ-oujda.ac.ma
Abdeluaab Lidouh
Université Mohammed premier, Faculté des sciences, Oujda, Maroc
E-mail address: lidouh@sciences.univ-oujda.ac.ma
Belkassem Seddoug
Université Mohammed premier, Faculté des sciences, Oujda, Maroc
E-mail address: seddougbelkassem@yahoo.fr