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research
An existence result for a nonlinear transmission problems
Authors
G. Mishuris
M. Dalla Riva
Publication date
22 January 2015
Publisher
View
on
arXiv
Abstract
Let
Ω
o
\Omega^o
Ω
o
and
Ω
i
\Omega^i
Ω
i
be open bounded subsets of
R
n
\mathbb{R}^n
R
n
of class
C
1
,
α
C^{1,\alpha}
C
1
,
α
such that the closure of
Ω
i
\Omega^i
Ω
i
is contained in
Ω
o
\Omega^o
Ω
o
. Let
f
o
f^o
f
o
be a function in
C
1
,
α
(
∂
Ω
o
)
C^{1,\alpha}(\partial\Omega^o)
C
1
,
α
(
∂
Ω
o
)
and let
F
F
F
and
G
G
G
be continuous functions from
∂
Ω
i
×
R
\partial\Omega^i\times\mathbb{R}
∂
Ω
i
×
R
to
R
\mathbb{R}
R
. By exploiting an argument based on potential theory and on the Leray-Schauder principle we show that under suitable and completely explicit conditions on
F
F
F
and
G
G
G
there exists at least one pair of continuous functions
(
u
o
,
u
i
)
(u^o, u^i)
(
u
o
,
u
i
)
such that
{
Δ
u
o
=
0
inÂ
Ω
o
∖
c
l
Ω
i
 
,
Δ
u
i
=
0
inÂ
Ω
i
 
,
u
o
(
x
)
=
f
o
(
x
)
for allÂ
x
∈
∂
Ω
o
 
,
u
o
(
x
)
=
F
(
x
,
u
i
(
x
)
)
for allÂ
x
∈
∂
Ω
i
 
,
ν
Ω
i
â‹…
∇
u
o
(
x
)
−
ν
Ω
i
â‹…
∇
u
i
(
x
)
=
G
(
x
,
u
i
(
x
)
)
for allÂ
x
∈
∂
Ω
i
 
,
\left\{ \begin{array}{ll} \Delta u^o=0&\text{in }\Omega^o\setminus\mathrm{cl}\Omega^i\,,\\ \Delta u^i=0&\text{in }\Omega^i\,,\\ u^o(x)=f^o(x)&\text{for all }x\in\partial\Omega^o\,,\\ u^o(x)=F(x,u^i(x))&\text{for all }x\in\partial\Omega^i\,,\\ \nu_{\Omega^i}\cdot\nabla u^o(x)-\nu_{\Omega^i}\cdot\nabla u^i(x)=G(x,u^i(x))&\text{for all }x\in\partial\Omega^i\,, \end{array} \right.
⎩
⎨
⎧
​
Δ
u
o
=
0
Δ
u
i
=
0
u
o
(
x
)
=
f
o
(
x
)
u
o
(
x
)
=
F
(
x
,
u
i
(
x
))
ν
Ω
i
​
â‹…
∇
u
o
(
x
)
−
ν
Ω
i
​
â‹…
∇
u
i
(
x
)
=
G
(
x
,
u
i
(
x
))
​
inÂ
Ω
o
∖
cl
Ω
i
,
inÂ
Ω
i
,
for allÂ
x
∈
∂
Ω
o
,
for allÂ
x
∈
∂
Ω
i
,
for allÂ
x
∈
∂
Ω
i
,
​
where the last equality is attained in certain weak sense. In a simple example we show that such a pair of functions
(
u
o
,
u
i
)
(u^o, u^i)
(
u
o
,
u
i
)
is in general neither unique nor local unique. If instead the fourth condition of the problem is obtained by a small nonlinear perturbation of a homogeneous linear condition, then we can prove the existence of at least one classical solution which is in addition locally unique
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