Nonsmooth analysis of doubly nonlinear evolution equations

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.Comment: 45 page

On systems of Cahn–Hilliard and Allen–Cahn equations considered as gradient flows in Hilbert spaces

“…It is then also much harder to prove applicability of the metric approach even in case (1.40) is satisfied by the functional S, as the following discussion will show. Thus, the scope of application is different from [28].…”

“…They also construct solutions for distances defined through Finstlerian structures. Another recent and conceptually related result is due to Mielke, Rossi and Savaré [28]: They considered a Finstler structure generalizing 1.1(2) but claiming convergence in (1.4) only for u n → u in H.…”

Lipschitz Regularity for a Homogeneous Doubly Nonlinear PDE

“…Doubly nonlinear equations such as (1.1) have mostly been studied from a functional analytic point of view. See for instance [MRS13] and [Ste08]. However, the pointwise properties and in particular the regularity theory has not been developed.…”

A doubly nonlinear evolution for the optimal Poincaré inequality

“…Using an implicit time scheme such as (1.5) to solve doubly nonlinear evolutions in reflexive Banach spaces has been carried out with great success; see [2,4,11,12,16,27,31]. In our view, the main insight that makes this approach work is a certain compactness feature of weak solutions that we now explore.…”