Tykhonov well-posedness of a mixed variational problem

Abstract

We consider a mixed variational problem governed by a nonlinear operator and a set of constraints. Existence, uniqueness and convergence results for this problem have already been obtained in the literature. In this current paper we complete these results by proving the well-posedness of the problem, in the sense of Tykhonov. To this end we introduce a family of approximating problems for which we state and prove various equivalence and convergence results. We illustrate these abstract results in the study of a frictionless contact model with elastic materials. The process is assumed to be static and the contact is with unilateral constraints. We derive a weak formulation of the model which is in the form of a mixed variational problem with unknowns being the displacement field and the Lagrange multiplier. Then, we prove various results on the corresponding mixed problem, including its well-posedness in the sense of Tykhonov, under various assumptions on the data. Finally, we provide mechanical interpretation of our results.

Inverse problems for constrained parabolic variational-hemivariational inequalities *

“…[46,61] as the relaxed monotonicity condition. Further, if J(p, •) is a convex function, then (8) holds with α J = 0 since it reduces to the monotonicity of the (convex) subdifferential.…”

“…Various classes of inverse problems have been treated in [20,30,33,34,42] for quasi-variational inequalities, and in [43] for quasihemivariational inequalities. The convergence results obtained in [8] for an elliptic mixed variational problem, and in [9] for an elliptic variational-hemivariational inequality can be also used to study optimization problems for such systems. Moreover, we refer to [27] where a hemivariational inequality technique has been applied to an obstacle problem, to [29] in which a unilateral indentation problem is treated in a form of a quasi-variational inequality, and to [26] for optimality conditions and numerical realization of inverse problems for a variational inequality.…”

Tykhonov triples and convergence results for hemivariational inequalities

“…This concept was generalized to variational inequalities in [15,16] and to hemivariational inequalities in [8]. References in the field include [1,13,27,28,31]. An extension of this concept in the study of generic problems in metric spaces was considered in our recent paper [26].…”

Well-Posedness of Minimization Problems in Contact Mechanics

“…It is worth mentioning that the first basic criteria for well-posedness of optimization problems in metric spaces were established by Furi and Vignoli in [13,14]. For more details on well-posedness for optimization problems, we refer the readers to [15][16][17]. Extension of the concept of well-posedness to variational inequalities, mixed variational inequalities and hemivariational inequalities can be found in [18][19][20][21][22][23][24][25][26], for instance.…”