Nonsmooth Analysis of Doubly Nonlinear Second-Order Evolution Equations with Non-Convex Energy Functionals

TL;DR

This paper develops a framework for proving the existence of strong solutions to complex second-order evolution equations with nonsmooth, non-convex energy functionals, using regularization and discretization techniques.

Contribution

It introduces a novel approach combining generalized Moreau--Yosida regularization and semi-implicit schemes to handle doubly multivalued, nonsmooth, and non-convex energies in second-order evolution equations.

Findings

Existence of strong solutions via regularization and discretization.

Energy-dissipation inequality established for solutions.

Applicability demonstrated through specific examples.

Abstract

We investigate the existence of strong solutions to a general class of doubly multivalued and nonlinear evolution equations of second order. The multivalued operators are generated by the subdifferential of nonsmooth potentials that live in different spaces, $U$ and $V$, where in general $U \nsubseteq V$ and $V \nsubseteq U$. The proof is based on the regularization of the dissipation potential using the generalized Moreau--Yosida regularization and a semi-implicit time-discretization scheme, which demonstrates the existence of strong solutions to the regularized problem. The existence of solutions to the original problem is then shown by letting the regularization parameter converge to zero. Furthermore, we establish an energy-dissipation inequality for the solution. We conclude with applications of this abstract theory.