Ammari H., Benkirane A., Touzani A. (editors) Recent Developments in Nonlinear Analysis

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120

IsoValue

0.000676368

0.00223902

0.00380167

0.00536432

0.00692697

0.00848962

0.0100523

0.0116149

0.0131776

0.0147402

0.0163029

0.0178655

0.0194282

0.0209908

0.0225535

0.0241161

0.0256788

0.0272414

0.0288041

0.0303667

uh 60x60

Fig. 5. Isovalues of u

for h = 1/30.

ﬁnite element approximations. R.A.I.R.O Numerical Analysis vol. 17, n

◦

385–395 (1983).

4. Ricardo H. Nochetto. Sharp L

∞

-Error Estimates for semilinear Elliptic Prob-

lems with free boundaries. Numer. Maths. 54, 243-255 (1988).

5. R. Glowinski, J. L. Lions, R. Tr´emoli`eres. Analyse num´erique des in´equations

variationnelles Tome1. Dunod, Paris (1976).

6. F. Brezzi, W. W. Hagar. Error estimate for the ﬁnite element solution of

variational inequality, part II. Numer. Math. 31, 1–16 (1978).

7. Slimane, L; Bendali, A; Laborde, P. Mixed formulations for a class of vari-

ational inequalities. M2AN Math. Model. Numer. Anal. 38 (2004), no. 1,

177–201.

8. Degueil A. R´esolution par une m´ethode d’´el´ements ﬁnis d’un probl`eme de

STEPHAN en terme de temp´erature et en teneur en mat´eriau non gel´e. Th`ese

`eme

cycle, Bordeaux (1977).

9. I. Ekland, R. Temam. Analyse convexe et probl`emes variationnels. Dunod,

Paris (1974).

10. Slimane, L. M´ethodes mixtes et traitement du verrouillage num´erique pour la

r´esolution des in´equations variationnelles. Th`ese de l’Institut National des

Sciences Appliqu´ees, TOULOUSE (2001).

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121

IsoValue

0.025

0.075

0.125

0.175

0.225

0.275

0.325

0.375

0.425

0.475

0.525

0.575

0.625

0.675

0.725

0.775

0.825

0.875

0.925

0.975

mu exact 120x120

Fig. 6. Isovalues of exact µ for h = 1/60.

11. P. G. Ciarlet. The Finite Element Method for Elliptic problems. Amsterdam:

North Holland, (1978).

12. P. A. Raviart, J. M. Thomas. Introduction l’analyse num´erique des ´equations

aux d´eriv´ees partielles. Masson, Paris (1992).

13. L. A. Caﬀarelli. The Obstacle Problem revisted. J. Fourier Anal. Appl. 4,

383–402 (1998).

14. F. Heicht, O. Pironneau, A. Le Hyaric, K. Ohtsuka. Freefem++. Lab-

oratoire Jacques-Louis Lions, Universit´e Pierre et Marie Curie, Paris,

(http://www.freefem.org/ﬀ++).

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122

IsoValue

0.0051434

0.0618342

0.118525

0.175216

0.231906

0.288597

0.345288

0.401979

0.458669

0.51536

0.572051

0.628742

0.685433

0.742123

0.798814

0.855505

0.912196

0.968886

1.02558

1.08227

muh 120x120

Fig. 7. Isovalues of µ

for h = 1/60.

H1-erroron u

0.05 0.1 0.15 0.2

0.005

0.01

0.015

Fig. 8. The H

(Ω)-error on u for diﬀerent values of h.

December 28, 2009 12:15 WSPC - Proceedings Trim Size: 9in x 6in recent

123

An obstacle problem via a sequence of penalized problems

E. Azroul and M. Rhoudaf

D´epartement de Math´ematiques et Informatique

Facult´e des Sciences Dhar-Mahraz

B.P 1796 Atlas F`es, Morocco

In this paper, we shall concern with the existence result of unilateral parabolic

degenerated problems associated to the equations of the form

∂u

∂t

+ A(u) = f in Q

where A is a classical Leray-Lions operator acting from the weighted Sobolev

space L

(0, T, W

1,p

(Ω, w)) into its dual L

(0, T, W

−1,p

(Ω, w

∗

)), while the

datum f is assumed in L

The proof is based on the penalty methods.

Keywords: Unilateral parabolic degenerate problem; Existence result; Penalty

methods.

1. Introduction

In this paper, we investigate the problem of existence of solutions of the

obstacle problems associated to the following initial-boundary value prob-

lem:

)







∂u

∂t

− div(a(x, t, u, ∇u)) = f in Q

= Ω ×(0, T )

u = 0 on Σ = ∂Ω × (0, T )

u(0) = u

in Ω,

where Ω is an open bounded subset of IR

, N ≥ 1, T > 0, and we have

set Q

the cylinder Ω × (0, T ) and Σ its lateral surface.

We assume that a(x, t, ξ) : Q

× IR

→ IR

is a Carath´eodry function

(i.e., measurable with respect to (x, t) and continuous with respect to ξ)

satisfying the hypotheses (H

) below.

The data are taken such that: f ∈ L

), u

∈ L

(Ω) and u

≥ 0.

More precisely, this paper deals with the existence of solution to the obstacle

December 28, 2009 12:15 WSPC - Proceedings Trim Size: 9in x 6in recent

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degenerated parabolic problem (P

) in the sense of entropy solution:

)











u ≥ ψ a.e. in Q

(u) ∈ L

(0, T, W

1,p

(Ω, w)), u ∈ C([0, T ], L

(Ω))

Ω

(u −ϕ)(τ) dx +

∂ϕ

∂t

(u −ϕ) dx dt

≤

(u −ϕ) dx dt +

Ω

− ϕ(x, 0)) dx

ϕ ∈ K

∩ L

∞

) ∩ C([0, T ], L

(Ω)) such that

∂ϕ

∂t

∈ L

(0, T, W

−1,p

(Ω, w

∗

)), ∀ k > 0

where S

(t) =

(s) ds, ψ ∈ L

∞

(Ω) ∩ W

1,p

(Ω, w) and K

= {u ∈

(0, T, W

1,p

(Ω, w)), u ≥ ψ a.e. in Q

The aim of our work is to investigate the relationship between the pos-

sibility to ﬁnd solutions of (P ) by approximating the singular data f and

with sequences of regular functions. More precisely letting {f



} and u



be a standard approximation of f and u

(that is f



→ f in L

(Q) and



→ u

in L

(Ω)), and considering the approximate problem:



)







∂u



∂t

− div(a(x, t, u



, ∇u



)) −



− ψ)

−

= f



in Q



= 0 on Σ



(0) = u



in Ω.

We study the possibility to ﬁnd a solution of (P

) as a limit of a subse-

quence {u



} of solutions of (P



The penalized term



−ψ)

−

introduced in (P



) play a crucial role

in the proof of our main result, in particular this term allows to prove that

the solution u of (P

) belongs in K

(that is u ≥ ψ).

A priori estimates of the truncations T



) are obtained in

(0, T, W

1,p

(Ω, w)). For the passage to the limit, we prove the strong

converge of the truncation of u



and the almost everywhere convergence

of ∇u



is proved. An example of operator model is,

A(u) = −div(|x|

|∇u|

p−2

∇u), r > 0.

In this context of parabolic problems, if w ≡ 1, existence results for (P

)

have been proved in

when f belongs to L

(0, T, W

−1,p

(Ω)) and u

is in

(Ω). The case where f belongs to L

) is investigated in [,

1920

] while

the case w 6= 1, is studied in [,

222

] where in the ﬁrst work the authors have

studied the variational case (f ∈ L

(0, T, W

−1,p

(Ω, w

∗

)) and in the second

work the L

) case is treated.

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125

Let us mention that in the literature of unilateral problems, the elliptic

case is more studied, while the study of parabolic case is poor.

2. Preliminaries and basic assumptions

Let Ω be a bounded open subset of IR

, p be a real number such that

1 < p < ∞ and w = {w

(x), 1 ≤ i ≤ N } be a vector of weight functions, i.e.,

every component w

(x) is a measurable function which is strictly positive

a.e. in Ω. Further, we suppose in all our considerations that, there exists

> max(N, p) such that w

−p

∈ L

loc

(Ω), (1)

and

−1

p−1

∈ L

loc

(Ω), (2)

for any 0 ≤ i ≤ N.

We denote by W

1,p

(Ω, w) the space of all real-valued functions u ∈

(Ω, w

) such that the derivatives in the sense of distributions fulﬁll

∂u

∂x

∈ L

(Ω, w

) for all i = 1, ..., N.

Which is a Banach space under the norm,

kuk

1,p,w

Ω

|u(x)|

dx +

i=1

Ω

∂u(x)

∂x

(x) dx

. (3)

The condition (1) implies that C

∞

(Ω) is a subset of W

1,p

(Ω, w) and con-

sequently, we can introduce the subspace W

1,p

(Ω, w) of W

1,p

(Ω, w) as the

closure of C

∞

(Ω) with respect to the norm (3). Moreover, the condition (2)

implies that W

1,p

(Ω, w) as well as W

1,p

(Ω, w) are reﬂexive Banach spaces.

We recall that the dual space of weighted Sobolev spaces W

1,p

(Ω, w) is

equivalent to W

−1,p

(Ω, w

∗

), where w

∗

= {w

∗

= w

1−p

, i = 0, ..., N} and

where p

is the conjugate of p, i.e., p

p−1

. For more details, we refer the

reader to.

Now we state the following assumptions:

Assumption (H

) For 2 ≤ p < ∞, we suppose that the expression

k|uk| =

i=1

Ω

∂u

∂x

(x) dx

(4)

December 28, 2009 12:15 WSPC - Proceedings Trim Size: 9in x 6in recent

126

is a norm on W

1,p

(Ω, w) which is equivalent to (3) and that there exists a

weight function σ on Ω such that,

σ ∈ L

(Ω) and σ

−1

∈ L

(Ω). (5)

We assume also the Hardy inequality,



Ω

|u(x)|

σ dx



≤ c

i=1

Ω

∂u

∂x

(x) dx

, (6)

holds for every u ∈ W

1,p

(Ω, w) with a constant c > 0 independent of u.

Moreover, the imbedding

1,p

(Ω, w) ,→ L

(Ω, σ) (7)

expressed by the inequality (6) is compact.

Note that (W

1,p

(Ω, w), k|.k|) is a uniformly convex ( and thus reﬂexive)

Banach space.

Remark 2.1. Assume that w

(x) ≡ 1 and there exists ν ∈



, +∞



∩

P −1

, +∞

such that

N −1

, w

−ν

∈ L

(Ω) for all i = 1, ..., N. (8)

Note that the assumptions (1) and (8) imply that,

k|uk| =

i=1

Ω

∂u

∂x

(x) dx

(9)

is a norm deﬁned on W

1,p

(Ω, w) and its equivalent to (3) and that, the

imbedding

1,p

(Ω, w) ,→,→ L

(Ω) (10)

is compact [ see,

pp 46].

Thus the hypotheses (H

) is satisﬁed for σ ≡ 1.

Assumption (H

)

a(x, t, s, ξ).ξ ≥ α

i=1

|ξ

, (11)

where c

(x, t) is a positive function in L

(Q), and α, β are strictly positive

constants.

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We recall that, for k > 1 and s in IR, the truncation is deﬁned as

(s) =

(

s if |s| ≤ k

|s|

if |s| > k.

3. Some technical lemmas

3.1. Some functional properties of time-regularization of a

function u

In order to deal with time derivative, we introduce a time molliﬁcation of a

function u belonging in some weighted Lebesgue space. Thus we deﬁne for

all µ ≥ 0 and all (x, t) ∈ Q

= µ

−∞

˜u(x, s)exp(µ(s −t)) ds where ˜u(x, s) = u(x, s)χ

(0,T )

(s).

Proposition 3.1.

1) If u ∈ L

, w

), then, u

is measurable in Q

∂u

∂t

= µ(u −u

) and



(x) dx dt



≤



|u|

(x) dx dt



i.e.,

)

≤ kuk

)

2) If u ∈ W

1,p

, w), then u

→ u in W

1,p

, w) as µ → +∞.

3) If u

→ u in W

1,p

, w), then (u

)

→ u

in W

1,p

, w).

3.2. Some weighted imbedding and compactness results

In this section, we establish some imbedding and compactness results in

weighted Sobolev Spaces which allow in particular to extend in the settings

of weighted Sobolev spaces, some trace results and the Aubin’s and Simon’s

results

Let V = W

1,p

(Ω, w), H = L

(Ω, σ) and let V

∗

= W

−1,p

(Ω, w

∗

), with

(2 ≤ p < ∞).

Let X = L

(0, T, V ). The dual space of X is X

∗

= L

(0, T, V

∗

) where

= 1 and denoting the space W

(0, T, V, H) = {v ∈ X : v

∈ X

∗

}

endowed with the norm

kuk

= kuk

+ ku

∗

, (12)

December 28, 2009 12:15 WSPC - Proceedings Trim Size: 9in x 6in recent

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which is a Banach space. Here u

stands for the generalized derivative of u,

i.e.,

(t)ϕ(t) dt = −

u(t)ϕ

(t) dt for all ϕ ∈ C

∞

(0, T ).

Lemma 3.1.

The Banach space H is an Hilbert space and its dual H

can be identiﬁed

with him self, i.e., H

' H.

Indeed, let

F : H × H → IR

(f, g) 7→

Ω

fgσ dx.

Remark that F is a symmetric bilinear form, which is also continuous and

deﬁned positively, since

Ω

fgσ dx =

Ω

fσ

gσ

dx ≤



Ω

|f|

σ dx





Ω

|g|

σ dx



Then, the Banach space H is an Hilbert space.

Finally by a standard argument, we can identiﬁed H with its dual H

i.e.,

' H.

Lemma 3.2.

The evolution triple V ⊆ H ⊆ V

∗

is veriﬁed.

Lemma 3.3.

Assume that,

∂u

∂t

= h

+ k

in D

(Ω),

where h

and k

are bounded respectively in L

(0, T, W

1,p

(Ω, w

∗

) and in

) .

If u

is bounded in L

(0, T, W

1,p

(Ω, w)), then u

→ u in L

loc

, σ).

Lemma 3.4.

Let g ∈ L

, γ) and let g

∈ L

, γ), with kg

,γ)

≤ c, 1 <

r < ∞. If g

(x) → g(x) a.e in Q

, then g

* g in L

, γ), where *

denotes weak convergence and γ is a weight function on Q

December 28, 2009 12:15 WSPC - Proceedings Trim Size: 9in x 6in recent

129

Lemma 3.5.

Assume that (H

) and (H

) are satisﬁed and let (u

) be a sequence in

(0, T, W

1,p

(Ω, w)) such that u

* u weakly in L

(0, T, W

1,p

(Ω, w)) and

[a(x, t, u

, ∇u

) − a(x, t, u

, ∇u)][∇u

− ∇u] dxdt → 0. (13)

Then, u

→ u in L

(0, T, W

1,p

(Ω, w)).

Lemma 3.6.

Let V ⊆ H ⊆ V

∗

be an evolution triple. Then the imbedding

(0, T, V, H) ,→ C([0, T ]), H)

is continuous .

3.3. Main results

Theorem 3.1. Let u

∈ L

(Ω) such that u

≥ 0. Assume that (H

) and

) hold true. Then there exists at last one solution u ∈ C([0, T ]; L

(Ω))

such that u(x, 0) = u

a.e. and for all τ ∈]0, T ],











(u) ∈ L

(0, T, W

1,p

(Ω, w)), u ≥ ψ a.e. in Ω

Ω

(u(τ) − ϕ(τ)) dx + h

∂ϕ

∂t

, T

(u −ϕ)i

a(x, t, u, ∇u)∇T

(u −ϕ) dx dt

≤

(u −ϕ) dx dt +

Ω

− ϕ(x, 0)) dx

∀ k > 0 and ∀ ϕ ∈ K

∩L

∞

(Q) such that

∂ϕ

∂t

∈ L

(0, T, W

−1,p

(Ω, w

∗

)),

where Q

= Ω×]0, τ[.

Proof.

Step 1: A priori estimates

Consider the approximate problem



)

(

∂u



∂t

− div(a(x, t, u



, ∇u



)) −



− ψ)

−

= f



∈ L

(0, T, W

1,p

(Ω, w)), u



(x, 0) = u



where f



→ f strongly L

(Q), u



→ u

strongly L

(Ω).

Thanks to,

there exists at least one solution of the problem (P

