DOI: 10.68381/jca05-3 ISSN: 0944-6532
An Existence Result for a Class of Non Convex Problems of the Calculus of Variations
Giulia Treu
We consider the functional
\int_\Omega [h(\gamma_K(\nabla u(x))) + u(x)]\,dx \quad u(x) \in W^{1,1}_0(\Omega)
∫
Ω
[
h
(
γ
K
(
∇
u
(
x
)
)
)
+
u
(
x
)
]
d
x
u
(
x
)
∈
W
0
1
,
1
(
Ω
)
where
\gamma_K
γ
K
is the gauge function of a convex set K and h : [0, ∞[ → [0, ∞] is a possibly non convex function. In the case K ⊂ ℝ² is a closed polytope and Ω ⊂ ℝ² is a bounded convex set we provide a sufficient condition for the existence of the minimum. Besides, as a corollary, we give conditions on Ω ⊂ ℝ² and f : ℝ² → [0, ∞] that are sufficient to the existence of a minimizer of
\int_\Omega [f(\nabla u(x)) + u(x)]\,dx \quad u(x) \in W^{1,1}_0(\Omega)
∫
Ω
[
f
(
∇
u
(
x
)
)
+
u
(
x
)
]
d
x
u
(
x
)
∈
W
0
1
,
1
(
Ω
)
.