Journal article
Regularity of a class of non-uniformly nonlinear elliptic equations
Calculus of variations and partial differential equations, Vol.55(5), pp.1-18
10/2016
DOI: 10.1007/s00526-016-1064-x
Abstract
In this paper we obtain the interior
$$C^{1,\alpha }$$
C
1
,
α
regularity of weak solutions for a class of non-uniformly nonlinear elliptic equations
$$\begin{aligned} \text {div} ~\! \left( a_1\left( \left| \nabla u \right| \right) \nabla u + a_2\left( \left| \nabla u \right| \right) \nabla u \right) =0, \end{aligned}$$
div
a
1
∇
u
∇
u
+
a
2
∇
u
∇
u
=
0
,
including the following special model
$$\begin{aligned} \text {div} ~\! \left( \left| \nabla u \right| ^{p-2} \nabla u + \left| \nabla u \right| ^{q-2} \nabla u \right) =0\quad \ \text{ for } \text{ any } \ p, q>1. \end{aligned}$$
div
∇
u
p
-
2
∇
u
+
∇
u
q
-
2
∇
u
=
0
for
any
p
,
q
>
1
.
These equations come from variational problems whose model energy functional is given by
$$\begin{aligned} \mathcal {P}(u, \Omega )=: \int _{\Omega } B^1\left( \left| \nabla u \right| \right) + B^{2}\left( \left| \nabla u \right| \right) dx, \end{aligned}$$
P
(
u
,
Ω
)
=
:
∫
Ω
B
1
∇
u
+
B
2
∇
u
d
x
,
where
$$\begin{aligned} B^k(t)=\int _0^t \tau a_k(\tau )~d\tau \quad \text{ for } \quad t\ge 0 \quad \text{ and }\quad k=1,2. \end{aligned}$$
B
k
(
t
)
=
∫
0
t
τ
a
k
(
τ
)
d
τ
for
t
≥
0
and
k
=
1
,
2
.
We remark that
$$\begin{aligned} B^k(t)= |t|^{\alpha _k} \log \big ( 1+|t|\big ) \quad \text{ for } ~~ ~~\alpha _k>1~~~ \text{ and }~~k=1,2 \end{aligned}$$
B
k
(
t
)
=
|
t
|
α
k
log
(
1
+
|
t
|
)
for
α
k
>
1
and
k
=
1
,
2
satisfy the given conditions in this work.
Details
- Title: Subtitle
- Regularity of a class of non-uniformly nonlinear elliptic equations
- Creators
- Lihe Wang - Department of Mathematics University of Iowa Iowa IA 52242 USAFengping Yao - Department of Mathematics Shanghai University Shanghai 200444 China
- Resource Type
- Journal article
- Publication Details
- Calculus of variations and partial differential equations, Vol.55(5), pp.1-18
- Publisher
- Springer Berlin Heidelberg
- DOI
- 10.1007/s00526-016-1064-x
- ISSN
- 0944-2669
- eISSN
- 1432-0835
- Language
- English
- Date published
- 10/2016
- Academic Unit
- Mathematics
- Record Identifier
- 9984083260102771
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