The Degenerate Oblique Derivative Problem for Elliptic and Parabolic Equations

This book deals with the tangential oblique derivative problem for second order linear and non-linear elliptic and parabolic operators. In a large survey a lot of the most interesting results obtained during the last 30 years are proposed. Historically, the problem was stated first by Poincar? when studying the tides, but the same problem arises in the theory of Brownian motion, too. The main difficulties in investigating this problem are due to the fact that at the points of tangency...
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This book deals with the tangential oblique derivative problem for second order linear and non-linear elliptic and parabolic operators. In a large survey a lot of the most interesting results obtained during the last 30 years are proposed. Historically, the problem was stated first by Poincar? when studying the tides, but the same problem arises in the theory of Brownian motion, too. The main difficulties in investigating this problem are due to the fact that at the points of tangency between the vector field, representing the boundary operator, and the boundary of the domain the Lopatinskii condition is failed and boundary value problems with infinite dimensional kernel or cokernel can appear. By using subelliptic type estimates for pseudodifferential operators in Sobolev and H?lder spaces many interesting results have been proved for linear problems during the last 30 years. The authors propose for the first time an investigation of the degenerate oblique derivative problem for semilinear elliptic and parabolic operators. To do this, they use subelliptic estimates (Egorov, H?rmander, Tr?ves, Winzell, Guan, Sawyer) and the Leray-Schauder fixed point principle. In this way theorems on existence, uniqueness and regularity of the classical solutions in H?lder classes are derived. In a lot of cases considered the coefficients are not infinitely smooth, and the set of degeneration of the problem is a rather massive one, i.e., it is not obliged to be a submanifold of the boundary and can have positive measure.