The Vishik--Lyusternik method in the mixed problem for parabolic operators unresolved with respect to the highest time derivative

We consider the mixed problem for parabolic operators unresolved with respect to the highest time derivative with boundary conditions of general type and zero initial conditions. We present an analog of the Shapiro--Lopatinskii condition that allows one to obtain two-sided \emph{a priori} estimates...

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Bibliographic Details
Published in:Transactions of the Moscow Mathematical Society 2007-11, Vol.68, p.67-92
Main Author: Volevich, L. R.
Format: Article
Language:English
Online Access:Get full text
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Summary:We consider the mixed problem for parabolic operators unresolved with respect to the highest time derivative with boundary conditions of general type and zero initial conditions. We present an analog of the Shapiro--Lopatinskii condition that allows one to obtain two-sided \emph{a priori} estimates in specially constructed function spaces. In the case considered in this paper the characteristic equation in the half-space has two groups of roots with different asymptotics. Because of this, the crucial role in the study of the problem is played by the Vishik--Lyusternik method in the form presented by Volevich (2006).
ISSN:0077-1554
1547-738X
DOI:10.1090/S0077-1554-07-00161-6