Structural stability of flows via evolutionary Gamma-convergence of weak-type

TL;DR

This paper develops a framework for analyzing the stability of multi-valued operator flows in Banach spaces using evolutionary Gamma-convergence of weak type, with applications to quasilinear PDEs and doubly-nonlinear flows.

Contribution

It introduces a novel notion of evolutionary Gamma-convergence of weak type and extends stability results to operator perturbations in Banach space flows.

Findings

Established a minimization principle for multi-valued operator problems.

Proved stability of these problems under data and operator perturbations.

Applied the theory to quasilinear PDEs, including doubly-nonlinear flows.

Abstract

The initial-value problem associated with multi-valued operators in Banach spaces is here reformulated as a minimization principle, extending results of Brezis-Ekeland, Nayroles and Fitzpatrick. At the focus there is the stability of these problems w.r.t.\ perturbations not only of data but also of operators; this is achieved via De Giorgi's $\Gamma$-convergence w.r.t.\ a nonlinear topology of weak type. A notion of evolutionary $\Gamma$-convergence of weak type is also introduced. These results are applied to quasilinear PDEs, including doubly-nonlinear flows.