Journal article
Singular perturbation method for inhomogeneous nonlinear free boundary problems
Calculus of variations and partial differential equations, Vol.49(3), pp.1237-1261
03/2014
DOI: 10.1007/s00526-013-0620-x
Abstract
In this paper, we study solutions of one phase inhomogeneous singular perturbation problems of the type:
$$ F(D^2u,x)=\beta _{\varepsilon }(u) + f_{\varepsilon }(x) $$
F
(
D
2
u
,
x
)
=
β
ε
(
u
)
+
f
ε
(
x
)
and
$$ \Delta _{p}u=\beta _{\varepsilon }(u) + f_{\varepsilon }(x)$$
Δ
p
u
=
β
ε
(
u
)
+
f
ε
(
x
)
, where
$$\beta _{\varepsilon }$$
β
ε
approaches Dirac
$$\delta _{0}$$
δ
0
as
$$\varepsilon \rightarrow 0$$
ε
→
0
and
$$f_{\varepsilon }$$
f
ε
has a uniform control in
$$L^{q}, q>N.$$
L
q
,
q
>
N
.
Uniform local Lipschitz regularity is obtained for these solutions. The existence theory for variational (minimizers) and non variational (least supersolutions) solutions for these problems is developed. Uniform linear growth rate with respect to the distance from the
$$\varepsilon -$$
ε
−
level surfaces are established for these variational and nonvaritional solutions. Finally, letting
$$\varepsilon \rightarrow 0$$
ε
→
0
basic properties such as local Lipschitz regularity and non-degeneracy property are proven for the limit and a Hausdorff measure estimate for its free boundary is obtained.
Details
- Title: Subtitle
- Singular perturbation method for inhomogeneous nonlinear free boundary problems
- Creators
- Diego Moreira - Departamento de Matemática UFC, Bloco 914, Campus do Pici Fortaleza Ceará 60455-760 BrazilLihe Wang - Department of Mathematics Shanghai Jiaotong University Shanghai 200240 China
- Resource Type
- Journal article
- Publication Details
- Calculus of variations and partial differential equations, Vol.49(3), pp.1237-1261
- Publisher
- Springer Berlin Heidelberg
- DOI
- 10.1007/s00526-013-0620-x
- ISSN
- 0944-2669
- eISSN
- 1432-0835
- Language
- English
- Date published
- 03/2014
- Academic Unit
- Mathematics
- Record Identifier
- 9984083254802771
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