© Hindawi Publishing Corp. REGULARIZATION OF A UNILATERAL OBSTACLE PROBLEM ON THE BOUNDARY

We give a regularization method for a unilateral obstacle problem with obstacle on the boundary and second member f.


Introduction.
Let Ω be a bounded domain of R n with smooth boundary Γ = ∂Ω.We consider the variational inequality problem-called obstacle problem: find where f ∈ L 2 (Ω); it is well known that problems (1.1) and (1.2) admit a unique solution (see [3]).The aim of this paper is to develop a regularization method for solving a nondifferentiable minimization problem that is equivalent to problems (1.1) and (1.2).
The idea of the regularization method is to approximate the nondifferentiable term by a sequence of differentiable ones depending on (ε ≥ 0, ε → 0).
We give three forms of regularization for which we establish the convergence result and a priori error estimates.
Next, by duality method of conjugate functions (see [1]), we provide a posteriori error estimates desired for the numerical computation.And as an application, we develop a regularization method for solving a sequence of penalised problem.

Formulation and regularization method.
Let Ω be a bounded domain of R n with smooth boundary Γ = ∂Ω and let f ∈ L 2 (Ω).
Consider the following variational inequality problem: where a(•, •) is defined by It is well known that problem (2.1) admit a unique solution (see [3]).For all z ∈ L 2 (Ω), we denote Let ϕ be the following functional: (2.5) The functional ϕ, being nondifferentiable on H 1 (Ω), is approximated by a sequence of differentiable functionals The regularized problem is (2.7) Problems (2.1) and (2.7), respectively, are equivalent to (2.8) (2.9) There are many methods to construct sequences of differentiable approximations.In this paper, we take the sequence φ ε verifying one of the following choices: (2.10)With these choices, problem (2.7) admits a unique solution.To establish the convergence of the sequence (u ε ), we need the following results (see [2]).
Lemma 2.2.Let V be a Hilbert space, a : V × V → R a continuous, V -elliptic bilinear, j : V → R proper, nonnegative, convex, weakly continuous function, and f a linear continuous on V .Assume that j ε : V → R, ε > 0, is a family of nonnegative convex weakly lower semicontinuous (l.s.c.) functions verifying (2.11) Let u and u ε be the solutions of the following variational inequalities: Taking v = u ε (resp., v = u) in the inequality of problem (2.1) (resp., (2.7)), we obtain (2.16) Consequently, we obtain the following a priori error estimates: (2.17)

A posteriori error estimates.
In this section, we use the duality method by conjugating functions in order to derive the a posteriori error estimates of the solution of approximate problem.We need the preliminary results (see [1]).
Let V and V * (resp., Y and Y * ) be two topological vector spaces and •, • V (resp., •, • Y ) denote the duality pairing between V and V * (resp., Y and Y * ).Let ϕ be a function from V to R = R∪{−∞, +∞}, and let its conjugate function be defined by where v * is in V * .Assume that there exists a continuous linear operator We consider the following minimization problem: where the conjugate function of J is given by Assume that V is a reflexive Banach space and Y a normed vector space.Let J : V ×Y → R be a proper l.s.c.strictly convex function verifying Then, problem (3.2) admits a unique solution, and Let Ω be an open subset of R n , and g : Ω × R n → R be the Carathéodory function, that is, for all s ∈ R n , x → g(x, s) is a measurable function, and, for almost all x ∈ Ω, the function s → g(x, s) is continuous.Then, the conjugate function of (assuming G is well defined over some function space V ) is where For problem (2.1), we take where y = (y 1 ,y 2 ) with y 1 ∈ (L 2 (Ω)) n and y 2 ∈ L 2 (Ω); a similar notation is used for y * ∈ Y * .So, the obstacle problem (2.1) can be rewritten in the form (3.2).
To apply Theorem 3.1, we compute the conjugate of the functional J.We have where (3.10) Hence, We have (3.12) Using (2.8) with v = u ε , we obtain (3.13) Applying Theorem 3.1 and using (3.11), we have Hence, u ε verifies the following Neumann problem: If we take then we have Then, we have the a posteriori error estimate Hence, we obtain the a posteriori error estimates.
For the choice (1.1) and (1.2), we have The a posteriori error estimate is For the choice (2.4), we have The a posteriori error estimate is 4. Application.Consider the following variational inequality problem: where a(•, •) is defined by It is well known that problem (4.1) admits a unique solution (see [3]).We write the obstacle problem (4.1) in a new form.
Proof (see [2]).The functional ϕ α , being nondifferentiable on H 1 (Ω), is approximated by a sequence of differentiable functionals The regularized problem is Problems (4.3) and (4.6) are, respectively, equivalent to (4.7) We can similarly proceed to drive the a posteriori error estimates; then, for the choice (1.1) and (1.2), we have ( The a posteriori error estimate is For the choice (2.4), the a posteriori error estimate is (4.10)

A posteriori error estimates for regularized discrete problem.
Let V h be a finite element space approximating H 1 (Ω).Then, the finite element solution u h ∈ V h for the obstacle problem (4.1) is determined from the following problem: (5.1) If we use the penalisation method, then the solution u h of problem (5.1) is the limit when α tends to 0 of the solution u α,h of the following problem: We can similarly proceed as in [2] to prove the convergence of the finite element approximations and to have a priori error estimates.
The regularized problem of (5.2) is We can similarly prove that (5.3) has unique solution and its solution converges to the corresponding solution of problem (5.2).By the duality theory on the discrete problems, we prove the following a posteriori error estimates.
For the choice (1.1) and (1.2), the a posteriori error estimate is (5.4) For the choice (2.4), the a posteriori error estimate is (5.5)

Mathematical Problems in Engineering
Special Issue on Time-Dependent Billiards

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space.Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon.Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles.His original model was then modified and considered under different approaches and using many versions.Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).We intend to publish in this special issue papers reporting research on time-dependent billiards.The topic includes both conservative and dissipative dynamics.Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned.Mathematical papers regarding the topics above are also welcome.