AbstractWe 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
).
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