A regularity result for minimal configurations of a free interface problem

We prove a regularity result for minimal configurations of variational problems involving both bulk and surface energies in some bounded open region $$\varOmega \subseteq {\mathbb {R}}^n$$ Ω ⊆ R n . We will deal with the energy functional $${\mathscr {F}}(v,E):=\int _\varOmega [F(\nabla v)+1_E G(\nabla v)+f_E(x,v)]\,dx+P(E,\varOmega )$$ F ( v , E ) : = ∫ Ω [ F ( ∇ v ) + 1 E G ( ∇ v ) + f E ( x , v ) ] d x + P ( E , Ω ) . The bulk energy depends on a function v and its gradient $$\nabla v$$ ∇ v . It consists in two strongly quasi-convex functions F and G, which have polinomial p-growth and are linked with their p-recession functions by a proximity condition, and a function $$f_E$$ f E , whose absolute valuesatisfies a q-growth condition from above. The surface penalization term is proportional to the perimeter of a subset E in $$\varOmega $$ Ω . The term $$f_E$$ f E is allowed to be negative, but an additional condition on the growth from below is needed to prove the existence of a minimal configuration of the problem associated with $${\mathscr {F}}$$ F . The same condition turns out to be crucial in the proof of the regularity result as well. If (u, A) is a minimal configuration, we prove that u is locally Hölder continuous and A is equivalent to an open set $${\tilde{A}}$$ A ~ . We finally get $$P(A,\varOmega )={\mathscr {H}}^{n-1}(\partial {\tilde{A}}\cap \varOmega $$ P ( A , Ω ) = H n - 1 ( ∂ A ~ ∩ Ω ).