ON THE SOLVABILITY OF A VARIATIONAL INEQUALITY PROBLEM AND APPLICATION TO A PROBLEM OF TWO MEMBRANES

The purpose of this work is to give a continuous convex function, for which we can characterize the subdifferential, in order to reformulate a variational inequality problem: find u = (u1,u2) ∈ K such that for all v = (v1,v2) ∈ K, ∫ Ω∇u1∇(v1 −u1)+ ∫ Ω∇u2∇(v2−u2)+(f ,v−u)≥ 0 as a system of independent equations, where f belongs to L2(Ω)×L2(Ω) and K = {v ∈H1 0(Ω)×H 0(Ω) : v1 ≥ v2 a.e. in Ω}. 2000 Mathematics Subject Classification. Primary 35J85.


Introduction.
We are interested in the following variational inequality problem: where f belongs to L 2 (Ω) × L 2 (Ω) and K is a closed convex set of H 1 0 (Ω) × H 1 0 (Ω) defined by Thanks to the orthogonal projection of the space L 2 (Ω) × L 2 (Ω) onto the cone defined by we construct a functional ϕ for which we can characterize the subdifferential at a point u, in order to reformulate problem (1.1) to a variational inequality without constraints; that is, find where ϕ is a continuous convex function from We prove that the solution u = (u 1 ,u 2 ) can be obtained as a solution of a system of independent two Dirichlet problems where g 1 and g 2 are two functions of L 2 (Ω) determined in terms of f 1 and f 2 .We will give an algorithm for computing these functions.
This approach can be applied to study a variational inequality arising from a problem of two membranes [2].

Formulation of the problem.
Let Ω be an open bounded set of R n with smooth boundary ∂Ω.We equip where For r ∈ L 2 (Ω), we let (2.4) the projection of v onto the cone given by (1.3) with respect to the scalar product of L 2 (Ω) × L 2 (Ω) (respectively, the projection with respect to the scalar product of L 2 (Ω) × L 2 (Ω) on the polar cone of defined by 0 = {v = (−r ,r ) ∈ L 2 (Ω) × L 2 (Ω) : r ≥ 0 a.e. on Ω}).We easily verify that We denote by (•, •) and • the scalar product and the norm of L 2 (Ω) × L 2 (Ω), respectively.We consider the following variational inequality problem: (2.9) It admits a unique solution.The functional ϕ defined from

9) if and only if u is the solution of the following problem: find
(2.10) It is well known in the general theory of variational inequalities that problem (2.10) admits a unique solution.So, it is sufficient to show that the solution u of (2.10) is an element of K. Let v = u + , then the inequality of (2.10) becomes (2.12) are the solution of (2.12), then by definition of µ ∈ ∂ϕ(u), we have Conversely, let u be the solution of problem (2.10).For v = u±w, with w ∈ H 1 0 (Ω)× H 1 0 (Ω), the inequality of (2.10) gives We deduce that a(u, w) We set (2.17)

.18)
We put v = 0, next v = 2u in (2.18).Since ϕ is positively homogeneous of degree 1, we obtain ϕ(u) = (µ, u) and consequently . For all v ∈ V , we have We deduce from Lemma 2.3 the following relations: (2.21) Indeed, the function ϕ being positively homogeneous of degree 1, µ ∈ ∂ϕ(u) implies , and taking into account Lemma 2.3, we can write problem (2.12) as follows: (2.24) Let A be the Riesz-Fréchet representation of Problem (2.24) can be written in the following form: (2.26) where P M (−t) is the projection of −t onto the closed convex set M with respect to the scalar product a(•, From the equality of Proposition 2.2, we deduce that the solution u of problem (2.9) verifies the following equations: (2.28) We notice that the prior knowledge of µ = (µ 1 ,µ 2 ) in terms of data of problem (2.9) yields the solutions u 1 and u 2 as solutions of two independent Dirichlet problems given by the system (2.28).We recall that for each element f of L p (Ω), the solution of the problem verifies the following properties (see [2]): where C is a constant depending only on p and Ω.We deduce from (2.28) that u 1 ,u 2 are in H 2 (Ω) and (2.31) where c, c 1 , and c 2 are constants depending only on Ω.We define the domain of noncoincidence [2] by

.33)
When u 1 and u 2 are continuous on Ω, the following relations are verified:

Application.
This method of solvability can be applied to the study of a variational inequality arising from a problem of two membranes [2], where Ω + and Ω − , are two parts of Ω (unknown) separated by a hypersurface Γ of R n such that Ω = Ω + ∪ Γ ∪ Ω − ; f 1 , f 2 are two regular functions and λ ∈ R. Formally, Ω + is the non-coincidence domain given by (2.32).