Affiliation:
1. Dipartimento di Scienze Fisiche, Informatiche e Matematiche , Università degli Studi di Modena e Reggio Emilia , via Campi 213/b, 41125 Modena , Italy
2. Dipartimento di Matematica e Informatica “U. Dini” , Università di Firenze , Viale Morgagni 67/A, 50134 Firenze , Italy
Abstract
Abstract
Integrals of the Calculus of Variations with
p
,
q
{p,q}
-growth may have not smooth minimizers, not even bounded, for general
p
,
q
{p,q}
exponents. In this paper we consider the scalar case, which contrary to the vector-valued one allows us not to impose structure conditions on the integrand
f
(
x
,
ξ
)
{f(x,\xi)}
with dependence on the modulus of the gradient, i.e.
f
(
x
,
ξ
)
=
g
(
x
,
|
ξ
|
)
{f(x,\xi)=g(x,|\xi|)}
. Without imposing structure conditions, we prove that if
q
p
{\frac{q}{p}}
is sufficiently close to 1, then every minimizer is locally Lipschitz-continuous.
Subject
Applied Mathematics,Analysis
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