Abstract
AbstractWe prove global $$W^{1,q}(\Omega ,{\mathbb {R}}^N)$$
W
1
,
q
(
Ω
,
R
N
)
-regularity for minimisers of $${{\mathscr {F}}(u)=\int _\Omega F(x,Du)\,{\mathrm{d}}x}$$
F
(
u
)
=
∫
Ω
F
(
x
,
D
u
)
d
x
satisfying $$u\ge \psi $$
u
≥
ψ
for a given Sobolev obstacle $$\psi $$
ψ
. $$W^{1,q}(\Omega ,{\mathbb {R}}^N)$$
W
1
,
q
(
Ω
,
R
N
)
regularity is also proven for minimisers of the associated relaxed functional. Our main assumptions on F(x, z) are a uniform $$\alpha $$
α
-Hölder continuity assumption in x and natural (p, q)-growth conditions in z with $$q<\frac{(n+\alpha )p}{n}$$
q
<
(
n
+
α
)
p
n
. In the autonomous case $$F\equiv F(z)$$
F
≡
F
(
z
)
we can improve the gap to $$q<\min \left( \frac{np}{n-1},p+1\right) $$
q
<
min
np
n
-
1
,
p
+
1
, a result new even in the unconstrained case.
Funder
Engineering and Physical Sciences Research Council
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Analysis
Cited by
10 articles.
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