Evolutionary quasi-variational and variational inequalities with constraints on the derivatives

TL;DR

This paper develops a general framework for analyzing the existence and uniqueness of solutions to nonlinear evolution systems with derivative constraints in Banach spaces, extending previous results to more general settings.

Contribution

It introduces a novel approach using double penalisation/regularisation to prove existence without coercivity, and extends uniqueness and continuous dependence results to broader classes.

Findings

Existence of weak solutions via double penalisation/regularisation.

Uniqueness and continuous dependence for time-independent convex sets.

Extension of previous results to more general nonlinear evolution systems.

Abstract

This paper considers a general framework for the study of the existence of quasi-variational and variational solutions to a class of nonlinear evolution systems in convex sets of Banach spaces describing constraints on a linear combination of partial derivatives of the solutions. The quasi-linear operators are of monotone type, but are not required to be coercive for the existence of weak solutions, which is obtained by a double penalisation/regularisation for the approximation of the solutions. In the case of time-dependent convex sets that are independent of the solution, we show also the uniqueness and the continuous dependence of the strong solutions of the variational inequalities, extending previous results to a more general framework.