Variational problems with non-constant gradient constraints

Abstract

This paper studies existence, uniqueness, continuous dependence on the given data and the asymptotic behavior of the solution of an evolutive variational inequality with non-constant gradient constraint and homogeneous Dirichlet boundary condition. With assumptions on the given data, we prove existence of solution for a variational inequality with two obstacles, a Lagrange multiplier problem and an equation with gradient constraint. Equivalence of these problems with the variational inequality with gradient constraint is proved. An example of non-equivalence among these problems is given in order to show the necessity of the assumptions.