Calculus of Variations and Geometric Measure Theory

L. Lamberti

A regularity result for minimal configurations of a free interface problem

created by lamberti on 31 Mar 2021
modified on 18 Feb 2022



Inserted: 31 mar 2021
Last Updated: 18 feb 2022

Pages: 15
Year: 2020


We prove a regularity result for minimal configurations of variational problems involving both bulk and surface energies in some bounded open region $\Omega \subseteq \mathbb{R}^n$. We will deal with the energy functional $\mathcal{F}(u,A):=\int_\Omega [F(\nabla u)+1_A G(\nabla u)+f_A(x,u)]\,dx+P(A,\Omega)$. The bulk energy depends on a function $u$ and its gradient $\nabla u$. It consists in two quasi-convex functions $F$ and $G$, which have polinomial $p$-growth and are $p$-homogeneous, and a function $f_A$, whose absolute value satisfies a $q$-growth condition from above. The surface penalization term is proportional to the perimeter of a subset $A$ in $\Omega$. The existence of a minimal configuration of the problem associated with $\mathcal{F}$ is ensured by an additional hypothesis we require on $f_A$ about its growth from below. If $(u,A)$ is a minimal configuration, we prove that $u$ is locally Hölder continuous and $A$ is equivalent to an open set $\tilde{A}$. We finally get $P(A,\Omega)=\mathcal{H}^{n-1}(\partial \tilde{A}\cap\Omega)$.

Keywords: regularity, free boundary problem, perimeter penalization, nonlinear variational problem