A nonlinear history-dependent boundary value problem

Abstract

We consider a nonlinear boundary value problem with unilateral constraints in a two-dimensional rectangle. We derive a variational formulation of the problem which is in the form of a history-dependent variational inequality. Then, we establish the existence of a unique weak solution to the problem. We also prove two convergence results. The first one provides the continuous dependence of the solution with respect to the unilateral constraint. The second one shows the convergence of the solution of the penalized problem to the solution of the original problem, as the penalization parameter converges to zero.