Submitted Paper

Inserted: 13 jul 2024
Last Updated: 22 aug 2024

Year: 2024

Abstract:

In this paper, we study the problem of minimizing the weighted total variation of a normalized $BV$ 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 ${{\Gamma^c}} \subset \partial\Omega$. We show that this problem is a relaxation of some shape optimization problem of type ${\it{Cheeger}}$, that is both problems have the same minimum. Then, we prove that the level sets of minimizers are optimal sets. Finally, we will also study the regularity as well as some properties of these optimal sets.

Keywords: shape optimization, 1-laplacian, Generalized Cheeger sets, Total variation minimization, Mixed Dirichlet-Neumann boundary conditions

Download:

• Weighted total variation minimization problem with mixed boundary conditions