Existence of functions of fixed trace and minimum weighted total variation

TL;DR

This paper proves the existence of minimizers for a weighted total variation functional with fixed boundary trace in smooth domains, using a constructive proof method.

Contribution

It establishes the existence of minimizers for a weighted total variation problem with fixed boundary conditions in smooth domains, advancing the understanding of such variational problems.

Findings

Existence of minimizers for the weighted total variation functional.

Constructive proof method for the existence result.

Applicable to smooth domains with fixed boundary trace.

Abstract

For \Omega a C^{2}-smooth domain, and a positive bounded continuous map a \in C(\Omega), we prove existence of a minimizer of the functional u \mapsto $\int_{\Omega} a|Du| over the space BV(\Omega) of functions of bounded variation with fixed trace f \in L^{1}(\partial\Omega). The method of proof is constructive.