Quasivariational solutions for first order quasilinear equations with gradient constraint

TL;DR

This paper establishes the existence of solutions for a class of evolution quasi-variational inequalities involving first order quasilinear operators with gradient constraints that depend on the solution, using regularization and a priori estimates.

Contribution

It introduces a novel approach to handle gradient constraints depending on the solution in quasi-variational inequalities, including existence and asymptotic behavior analysis.

Findings

Existence of solutions for the evolution quasi-variational inequality.

Existence of stationary solutions and their asymptotic behavior.

Uniqueness results in the variational case with solution-independent constraints.

Abstract

We prove the existence of solutions for an evolution quasi-variational inequality with a first order quasilinear operator and a variable convex set, which is characterized by a constraint on the absolute value of the gradient that depends on the solution itself. The only required assumption on the nonlinearity of this constraint is its continuity and positivity. The method relies on an appropriate parabolic regularization and suitable {\em a priori} estimates. We obtain also the existence of stationary solutions, by studying the asymptotic behaviour in time. In the variational case, corresponding to a constraint independent of the solution, we also give uniqueness results.