Calculus of Variations and Geometric Measure Theory

E. De Giorgi - M. Carriero - A. Leaci

Existence theorem for a minimum problem with free discontinuity set

created by leaci on 15 Apr 2023


Published Paper

Inserted: 15 apr 2023
Last Updated: 15 apr 2023

Journal: Archive for Rational Mechanics and Analysis
Volume: 108
Pages: 195-218
Year: 1989
Links: Published paper


We study the variational problem

$\displaystyle \min \Bigl\{ \int_{\Omega\setminus K} \vert\nabla u\vert^2 dx + \mu \int_{\Omega\setminus K}\vert u-g\vert^q dx + \lambda H_{n-1}(K \cap \Omega): $

$\qquad\qquad K \subset {\mathbb R}^n \hbox{ closed set},\ u\in C^1(\Omega \setminus K)\Bigr\}, $

where $\Omega$ is an open set in ${\mathbb R}^n$,$n\ge 2$, $g\in L^q (\Omega) \cap L^\infty(\Omega)$, $1\le q<+\infty$, $0<\lambda, \mu<+\infty$

and $\mathcal{H}^{n−1}$ is the $(n−1)$-dimensional Hausdorff Measure.

Keywords: calculus of variations, regularity, free discontinuity problems