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Linear parabolic equations with strong boundary degeneration. (English) Zbl 1443.35083

In this paper, the author considers linear second order parabolic diferential equations which are not uniformly parabolic but degenerate near (some part of) the boundary. In the main body of this work, such equations are studied in the framework of Riemannian manifolds, even if, for the sake of simplicity, it is analysed in detail only the simpler Euclidean setting. More precisely, as an application of the theory of linear parabolic diferential equations on noncompact Riemannian manifolds, developed in earlier papers by the same author, it is proved a maximal regularity theorem for nonuniformly parabolic boundary value problems in Euclidean spaces. The new feature of this result is the fact that, besides the already important goal of obtaining an optimal solution theory, it is considered the ‘natural’ case where the degeneration occurs only in the normal direction.

MSC:

35K65 Degenerate parabolic equations
35K45 Initial value problems for second-order parabolic systems
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
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