Weighted total variation minimization problem with mixed Dirichlet–Neumann boundary conditions

Samer Dweik

Vol. 335 (2025), No. 1, 53–79

DOI: 10.2140/pjm.2025.335.53

Abstract

We study the problem of minimizing the weighted total variation of a normalized $\mathrm{BV}<!--FUNCTION\; APPLICATION-->$ function $u$ plus a penalization on the weighted $L^1$ norm of the trace of $u$ on the Neumann part $\Gamma$ of the boundary, while assuming a Dirichlet condition $u=0$ on the complement part ${\Gamma }^c\subset \partial \mathrm{\Omega }$. We show that this problem is a relaxation of some shape optimization problem of type Cheeger, that is, both problems have the same minimum. Then, we prove that the level sets of minimizers are optimal sets. Finally, we study the regularity as well as some properties of these optimal sets.

Keywords

Cheeger problem, mixed boundary conditions, shape optimization

Mathematical Subject Classification

Primary: 35J20, 49J40, 49Q10

Milestones

Received: 23 March 2024

Revised: 12 November 2024

Accepted: 19 January 2025

Published: 24 March 2025