Affiliation:
1. DISIM , University of L’Aquila , Via Vetoio snc, Coppito, 67100 L’Aquila , Italy
Abstract
Abstract
We prove the absence of the Lavrentiev gap for non-autonomous functionals
ℱ
(
u
)
≔
∫
Ω
f
(
x
,
D
u
(
x
)
)
𝑑
x
,
\mathcal{F}(u)\coloneqq\int_{\Omega}f(x,Du(x))\,dx,
where the density
f
(
x
,
z
)
{f(x,z)}
is α-Hölder continuous with respect to
x
∈
Ω
⊂
ℝ
n
{x\in\Omega\subset\mathbb{R}^{n}}
, it satisfies the
(
p
,
q
)
{(p,q)}
-growth conditions
|
z
|
p
⩽
f
(
x
,
z
)
⩽
L
(
1
+
|
z
|
q
)
,
\lvert z\rvert^{p}\leqslant f(x,z)\leqslant L(1+\lvert z\rvert^{q}),
where
1
<
p
<
q
<
p
(
n
+
α
n
)
{1<p<q<p(\frac{n+\alpha}{n})}
, and it can be approximated from below by suitable densities
f
k
{f_{k}}
.
Subject
Applied Mathematics,Analysis
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1
<
p
<
2
{1
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New examples on Lavrentiev gap using fractals,
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Regularity for general functionals with double phase,
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