Nonsmooth analysis of doubly nonlinear evolution equations

TL;DR

This paper develops a framework for analyzing a broad class of nonsmooth, nonconvex doubly nonlinear evolution equations in Banach spaces, establishing existence results and applying them to a material model in elasticity.

Contribution

It introduces general conditions for existence of solutions to nonsmooth doubly nonlinear equations and employs variational techniques with time-discretization for proofs.

Findings

Established existence of solutions under broad conditions.

Applied the theory to a finite-strain elasticity model.

Used variational methods and limit processes in the analysis.

Abstract

In this paper we analyze a broad class of abstract doubly nonlinear evolution equations in Banach spaces, driven by nonsmooth and nonconvex energies. We provide some general sufficient conditions, on the dissipation potential and the energy functional,for existence of solutions to the related Cauchy problem. We prove our main existence result by passing to the limit in a time-discretization scheme with variational techniques. Finally, we discuss an application to a material model in finite-strain elasticity.