Prox-regularization and solution of ill-posed elliptic variational inequalities

Proximal point method and elliptic regularization

“…a Lipschitz continuous boundary) and the second Korn inequality, the existence of an operatorB satisfying condition (iii) of Theorem 1 can be shown, in particular, for ill-posed elliptic variational inequalities, which describe the problem of linear elasticity with given friction and the two-body contact problem (see [13] for the mathematical formulations and [5] for the proximal method with weak regularization). We deal with these problems in Section 4.3.…”

“…In [4,5] the proximal point method was developed for solving variational inequalities in elasticity theory: two-body contact problems without friction and static problems of linear elasticity with given friction have been investigated.…”

“…approximation of K and ∂ϕ is improved after each proximal iteration. If the operator Q possesses a certain reserve of monotonicity as described by the assumptions (1-ii) and (2-ii) below, conditions on the choice of the regularizing functional admit the application of weak proximal regularization as well as of regularization on a subspace (see [4,5] for these approaches).…”

Interior Proximal Method for Variational Inequalities: Case of Nonparamonotone Operators

“…This situation meets, for instance, in problems of linear elasticity with friction [7,14]. (H, · H ) be a Hilbert space such that X is dense and continuously embedded into H ; G: X → X be a linear monotone operator with the symmetry property…”

Some Results About Proximal-Like Methods

“…In [19] the convergence of MSR-methods is investigated for two elliptic variational inequalities in elasticity theory: the two body contact problem (without friction) and the Signorini problem. In both cases the approximation of K is performed by the FEM and the approximation of the multi-valued operator Q in the Signorini problem is like (9).…”