*Preprint*

**Inserted:** 31 mar 2021

**Last Updated:** 31 mar 2021

**Pages:** 15

**Year:** 2020

**Abstract:**

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 HÃ¶lder 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

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