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Boundary value problems for the Laplace equation in Lipschitzian domains. (English) Zbl 0608.31001

Recent progress in Fourier analysis, Proc. Semin., El Escorial/Spain 1983, North-Holland Math. Stud. 111, 33-48 (1985).
[For the entire collection see Zbl 0581.00009.]
In a bounded n-dimensional domain D whose boundary is locally the graph of a lipschitzian function, the authors consider for a harmonic function G the Dirichlet problem \(G|_{\partial D}=g\) and the oblique derivative problem \((\nabla G\cdot v)|_{\partial D}=g\) where v is prescribed continuous unit vector valued function on \(\partial D\) (the Neumann problem is not covered). They show that the Dirichlet problem is always uniquely solvable if the boundary data belong to \(L^ p\) for all p, \(p_ 0<p\leq 2\), where \(p_ 0\), \(1\leq p_ 0<2\), depends on the domain, and the oblique derivative problem is solvable with finitely many linear conditions imposed on the boundary data if the normal component of v has a positive lower bound. The boundary conditions are to be satisfied in a certain sense.
Reviewer: G.Jaiani

MSC:

31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
31B20 Boundary value and inverse problems for harmonic functions in higher dimensions
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation

Citations:

Zbl 0581.00009