Calculus of Variations and Geometric Measure Theory

M. Carriero - A. Leaci

Existence theorem for a Dirichlet problem with free discontinuity set

created by leaci on 15 Apr 2023


Published Paper

Inserted: 15 apr 2023
Last Updated: 15 apr 2023

Journal: Nonlinear Analysis: Theory, Methods & Applications
Volume: 15
Pages: 661-677
Year: 1990
Links: Published paper


We study the free discontinuity problem

$\displaystyle \min \left\{\int_{\Omega\setminus K}\vert\nabla u\vert^2 dx+\lambda{\cal H}\,_{n-1}(K) \right\}$

where the minimum is taken over all the closed sets $K\subset\overline{\Omega}$ and the functions $u\in C^1(\Omega\setminus K)\cap C^0(\overline{\Omega}\setminus (M\cup K))$

with $u=w\ {\rm on}\ \partial\Omega\setminus (M\cup K)$; here $\Omega$ is a bounded domain in ${\mathbb R}^{n}$, $n\ge 2$, such that ${\cal H}\,_{n-1}(\partial\Omega )<+\infty$

and $\partial\Omega$ is a $C^1$ surface up to an ${\cal H}\,_{n-1}$ negligible closed set $M$, $w\in C^1(\partial\Omega\setminus M)\cap L^\infty(\partial\Omega\setminus M)$,

$0<\lambda <+\infty$ and ${\cal H}\,_{n-1}$ is the $(n-1)$-dimensional Hausdorff measure.

Keywords: calculus of variations, free discontinuity problems