Sub - Elliptic Boundary Value Problems Forquasilinear Elliptic

Abstract

Classical solvability and uniqueness in the HH older space C 2+ () is proved for the oblique derivative problem a ij (x)D ij u + b(x; u; Du) = 0 in ; @u=@`= '(x) on @ in the case when the vector eld`(x) = (` 1 (x); : : : ; ` n (x)) is tangential to the boundary @ at the points of some non-empty set S @; and the nonlinear term b(x; u; Du) grows quadratically with respect to the gradient Du. 0. Introduction The paper is devoted to the study of so-called oblique derivative problem rstly posed by H. Poincar e ((Poi]): given a domain , nd a solution in of an elliptic diierential equation that satisses boundary condition in terms of directional derivative with respect to a vector eld`deened on the boundary @. More precisely, we shall be concerned with the problem (a ij (x)D ij u + b(x; u; Du) = 0 in ; @u=@`` i (x)D i u = '(x) on @ (0.1) in the degenerate (tangential) case, i.e. a situation when the vector eld`(x) = (` 1 (x); : : : ; ` n (x)) prescribing the boundary operator becomes tangential to @ at the points of some non-empty set S: This way, the well-known Shapiro{Lopatinskii complementary condition is violated on the set S and the classical theory (cf. G-T]) cannot be applied to the problem (0.1). The linear tangential problem (b(x; z; p) = b i (x)p i + c(x)z) has been very well studied in the last three decades. The pioneering works of Bicadze B] and HH or-mander H] indicated how the solvability and uniqueness properties depend on the way in which the normal component of`(x) changes its sign across S: More precisely, suppose S to be a submanifold of @ of co-dimension one, and let`(x) = (x)+(x)(x): Here (x) is the unit outward normal to @ and (x) is a tangential eld to @ such that j`(x)j = 1: There are three possible behaviors of`(x) near the set S = fx 2 @: (x) = 0g : a) `(x) is of neutral type: (x) 0 or (x) 0 on @; b) `(x) is of emergent type: the sign of (x) changes from ? to + in the positive direction on-integral curves through the points of S; c) `(x) is of submergent type: the sign of (x) changes from + to …

Cite this paper

@inproceedings{Palagachev1997SubE, title={Sub - Elliptic Boundary Value Problems Forquasilinear Elliptic}, author={Dian K. Palagachev and Peter Popivanov}, year={1997} }