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Higher regularity for nonlinear oblique derivative problems in Lipschitz domains

Gary M. Lieberman — 2002

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

There is a long history of studying nonlinear boundary value problems for elliptic differential equations in a domain with sufficiently smooth boundary. In this paper, we show that the gradient of the solution of such a problem is continuous when a directional derivative is prescribed on the boundary of a Lipschitz domain for a large class of nonlinear equations under weak conditions on the data of the problem. The class of equations includes linear equations with fairly rough coefficients as well...

Nonuniqueness for some linear oblique derivative problems for elliptic equations

Gary M. Lieberman — 1999

Commentationes Mathematicae Universitatis Carolinae

It is well-known that the “standard” oblique derivative problem, Δ u = 0 in Ω , u / ν - u = 0 on Ω ( ν is the unit inner normal) has a unique solution even when the boundary condition is not assumed to hold on the entire boundary. When the boundary condition is modified to satisfy an obliqueness condition, the behavior at a single boundary point can change the uniqueness result. We give two simple examples to demonstrate what can happen.

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