Non-diffusive Variational Problems with Distributional and Weak Gradient Constraints

TL;DR

This paper studies non-diffusive variational problems with gradient constraints, establishing existence, uniqueness, duality, and proposing a finite-element method with numerical validation.

Contribution

It introduces a comprehensive analysis of variational problems with distributional and weak gradient constraints, including duality theory and a novel finite-element approach.

Findings

Existence and uniqueness under low regularity assumptions

Identification of the Fenchel pre-dual problem

Numerical validation of the theoretical results

Abstract

In this paper, we consider non-diffusive variational problems with mixed boundary conditions and (distributional and weak) gradient constraints. The upper bound in the constraint is either a function or a Borel measure, leading to the state space being a Sobolev one or the space of functions of bounded variation. We address existence and uniqueness of the model under low regularity assumptions, and rigorously identify its Fenchel pre-dual problem. The latter in some cases is posed on a non-standard space of Borel measures with square integrable divergences. We also establish existence and uniqueness of solutions to this pre-dual problem under some assumptions. We conclude the paper by introducing a mixed finite-element method to solve the primal-dual system. The numerical examples confirm our theoretical findings.