Variational Problems with Non-Constant Gradient Constraints

Lisa Santos

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.

Keywords: Parabolic variational inequality; Parabolic PDE; Free boundary problem.

Classification (MSC2000): 35K85, 35K55, 35R35.

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