Variational and Quasi-Variational Inequalities with Gradient Type Constraints

TL;DR

This survey reviews stationary and evolutionary variational inequalities with gradient constraints, highlighting recent developments, properties, and new results relevant to mechanics and physics applications.

Contribution

It compiles known properties and introduces new results in variational and quasi-variational inequalities with gradient constraints.

Findings

Collected and presented several known properties.

Found a few novel results and examples.

Applied to models in mechanics and physics.

Abstract

This survey on stationary and evolutionary problems with gradient constraints is based on developments of monotonicity and compactness methods applied to large classes of scalar and vectorial solutions to variational and quasi-variational inequalities. Motivated by models for critical state problems and applications to free boundary problems in Mechanics and in Physics, in this work several known properties are collected and presented and a few novel results and examples are found.