Abstract
We establish some higher differentiability results of integer and fractional order for solutions to non-autonomous obstacle problems of the form
min
{
∫
Ω
f
(
x
,
D
v
(
x
)
)
:
v
∈
K
ψ
(
Ω
)
}
,
\min\biggl{\{}\int_{\Omega}f(x,Dv(x)):v\in\mathcal{K}_{\psi}(\Omega)\biggr{\}},
where the function 𝑓 satisfies 𝑝-growth conditions with respect to the gradient variable, for
1
<
p
<
2
1<p<2
, and
K
ψ
(
Ω
)
\mathcal{K}_{\psi}(\Omega)
is the class of admissible functions
v
∈
u
0
+
W
0
1
,
p
(
Ω
)
v\in u_{0}+W^{1,p}_{0}(\Omega)
such that
v
≥
ψ
v\geq\psi
a.e. in Ω, where
u
0
∈
W
1
,
p
(
Ω
)
u_{0}\in W^{1,p}(\Omega)
is a fixed boundary datum.
Here we show that a Sobolev or Besov–Lipschitz regularity assumption on the gradient of the obstacle 𝜓 transfers to the gradient of the solution, provided the partial map
x
↦
D
ξ
f
(
x
,
ξ
)
x\mapsto D_{\xi}f(x,\xi)
belongs to a suitable Sobolev or Besov space.
The novelty here is that we deal with sub-quadratic growth conditions with respect to the gradient variable, i.e.
f
(
x
,
ξ
)
≈
a
(
x
)
|
ξ
|
p
f(x,\xi)\approx a(x)\lvert\xi\rvert^{p}
with
1
<
p
<
2
1<p<2
, and where the map 𝑎 belongs to a Sobolev or Besov–Lipschitz space.
Obstacle Problems and Maximal Operators
