Multivalued nonmonotone dynamic boundary condition

TL;DR

This paper introduces a new class of dynamic boundary hemivariational inequalities involving the Laplace operator with Wentzell boundary conditions and multivalued nonmonotone operators, providing existence and approximation results.

Contribution

It formulates and analyzes dynamic boundary hemivariational inequalities with multivalued nonmonotone operators, extending the theory and solution techniques for such problems.

Findings

Existence of weak solutions under growth and sign conditions.

Reformulation allows standard techniques to be applied.

Convergence of Faedo-Galerkin approximation sequences.

Abstract

In this paper, we introduce a new class of hemivariational inequalities, called dynamic boundary hemivariational inequalities reflecting the fact that the governing operator is also active on the boundary. In our context, it concerns the Laplace operator with Wentzell (dynamic) boundary conditions perturbed by a multivalued nonmonotone operator expressed in terms of Clarke subdifferentials. We will show that one can reformulate the problem so that standard techniques can be applied. We will use the well-established theory of boundary hemivariational inequalities to prove that under growth and general sign conditions, the dynamic boundary hemivariational inequality admits a weak solution. Moreover, in the situation where the functionals are expressed in terms of locally bounded integrands, a "filling in the gaps" procedure at the discontinuity points is used to characterize the subdifferential on the product space. Finally, we prove that, under a growth condition and eventually smallness conditions, Faedo-Galerkin approximation sequence converges to a desired solution.