Abstract
The purpose of this paper is twofold. Firstly we consider nonlinear nonsmooth elliptic boundary value problems, and also related parabolic initial boundary value problems that model in a simplified way steady-state unilateral contact with Tresca friction in solid mechanics, respectively, stem from nonlinear transient heat conduction with unilateral boundary conditions. Here a recent duality approach, that augments the classical Babuška-Brezzi saddle point formulation for mixed variational problems to twofold saddle point formulations, is extended to the nonsmooth problems under consideration. This approach leads to variational inequalities of mixed form for three coupled fields as unknowns and to related differential mixed variational inequalities in the time-dependent case. Secondly we are concerned with the stability of the solution set of a general class of differential mixed variational inequalities. Here we present a novel upper set convergence result with respect to perturbations in the data, including perturbations of the associated nonlinear maps, the nonsmooth convex functionals, and the convex constraint set. We employ epiconvergence for the convergence of the functionals and Mosco convergence for set convergence. We impose weak convergence assumptions on the perturbed maps using the monotonicity method of Browder and Minty.
Highlights
The classical Babuska-Brezzi theory for mixed variational problems has been extended by Gatica [1, 2] to some classes of variational problems and nonlinear operator equations
In this paper, we are concerned with stability of the solution set to differential mixed variational inequalities (DMVI)
Let us refer to [10], where a Lyapunov approach is developed for strong solutions of evolution variational inequalities and to [11], where first several sensitivity results are established for initial value problems of ordinary differential equations with nonsmooth right hand sides and applied to treat differential variational inequalities
Summary
The classical Babuska-Brezzi theory for mixed variational problems has been extended by Gatica [1, 2] to some classes of variational problems and nonlinear operator equations. In this paper, we are concerned with stability of the solution set to DMVI In this connection, let us refer to [10], where a Lyapunov approach is developed for strong solutions of evolution variational inequalities and to [11], where first several sensitivity results are established for initial value problems of ordinary differential equations with nonsmooth right hand sides and applied to treat differential variational inequalities.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have