Non-diffusive Variational Problems with Distributional and Weak Gradient Constraints
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.
Cite
@article{arxiv.2106.12680,
title = {Non-diffusive Variational Problems with Distributional and Weak Gradient Constraints},
author = {Harbir Antil and Rafael Arndt and Carlos N. Rautenberg and Deepanshu Verma},
journal= {arXiv preprint arXiv:2106.12680},
year = {2021}
}