Three-field Modelling of Nonsmooth Problems and Stability of Differential Mixed Variational Inequalities

Joachim Gwinner^*

The purpose of this contribution is two-fold. 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 two-fold saddle point formulations, is extended to the non-smooth 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, of the non-smooth convex functionals, and the convex constraint set.

Mathematics Subject Classification: 49J40 35J87 35K86

Keywords: unilateral contact; Tresca friction; nonlinear transient heat conduction; unilateral boundary conditions

Minisymposion: Nonsmooth and Unilateral Problems - Modelling, Analysis and Optimization Methods