singular perturbation
asymptotic approximation
boundary layer
implicit function theorem
space depending small diffusion coefficient
Boundary Layer Solutions to Problems
with Infinite Dimensional Singular and Regular Perturbations
Lutz Recke
Recke
Lutz
Oleh Omel'chenko
Omel'chenko
Oleh
Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976),
Boundary Layer Solutions to Problems
with Infinite Dimensional Singular and Regular Perturbations
Lutz Recke
,
Oleh Omel'chenko
Preprint series:
Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976),
MSC 2000
- 34E15 Singular perturbations, general theory
-
34B15 Nonlinear boundary value problems
-
47J07 Abstract inverse mapping and implicit function theorems
-
58C15 Implicit function theorems; global Newton methods
Abstract
We prove existence, local uniqueness
and asymptotic estimates for boundary layer solutions
to singularly perturbed equations of the type
$(\varepsilon(x)^2 u'(x))'=f(x,u(x))+
g(x,u(x),\varepsilon(x) u'(x)), 0< x <1$,
with Dirichlet and Neumann boundary conditions.
Here the functions~$\varepsilon$ and~$g$
are small and, hence, regarded
as singular and regular functional perturbation parameters.
The main tool of the proofs is a generalization
(to Banach space bundles) of an Implicit Function Theorem
of R. Magnus.
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