The Vishik--Lyusternik method in elliptic problems with a small parameter

We consider boundary value problems where the operator defined in a domain and the boundary operators depend on a small parameter. Elliptic and properly elliptic problems with a small parameter are defined. It is proved that small parameter ellipticity is a necessary and sufficient condition for the...

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Bibliographic Details
Published in:Transactions of the Moscow Mathematical Society 2006-12, Vol.67, p.87-125
Main Author: Volevich, L. R.
Format: Article
Language:English
Online Access:Get full text
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Summary:We consider boundary value problems where the operator defined in a domain and the boundary operators depend on a small parameter. Elliptic and properly elliptic problems with a small parameter are defined. It is proved that small parameter ellipticity is a necessary and sufficient condition for the existence of \emph{a priori} estimates that are uniform with respect to the parameter. The proof of uniform estimates is based on the construction of the exponential boundary layer introduced in the classical paper by Vishik and Lyusternik.
ISSN:0077-1554
1547-738X
DOI:10.1090/S0077-1554-06-00154-3