Autonomous Evolutionary Inclusions with Applications to Problems with Nonlinear Boundary Conditions

TL;DR

This paper develops a theoretical framework for autonomous differential inclusions in Hilbert spaces, establishing well-posedness and causality, and applies it to nonlinear boundary value problems with specific boundary conditions.

Contribution

It introduces a class of autonomous differential inclusions with a proof of well-posedness and causality, covering nonlinear boundary value problems with new boundary operator conditions.

Findings

Operators are shown to be maximal monotone in time and space.

Certain nonlinear boundary value problems are encompassed by this class.

Two examples illustrate the applicability of the theory.

Abstract

We study an abstract class of autonomous differential inclusions in Hilbert spaces and show the well-posedness and causality, by establishing the operators involved as maximal monotone operators in time and space. Then the proof of the well-posedness relies on a well-known perturbation result for maximal monotone operators. Moreover, we show that certain types of nonlinear boundary value problems are covered by this class of inclusions and we derive necessary conditions on the operators on the boundary in order to apply the solution theory. We exemplify our findings by two examples.