On the oblique derivative problem for harmonic functions
type: Item
instance of: scholarly article
title: On the oblique derivative problem for harmonic functions
zbMATH Open document ID: 0964.31004
DOI: 10.1007/BF02465817
published in: Lithuanian Mathematical Journal
publication date: 2001-07-22T00:00:00Z
review text: For a bounded domain \(D\) with smooth boundary \(\Gamma\), the oblique derivative problem is to find a harmonic function \(u\) regular in \(D\), continuously differentiable in \(D\cup \Gamma\), and satisfying the condition \(a \frac{\partial u}{\partial n}+ l(u)= b\) where \(a\) and \(b\) are given differentiable functions and \(l\) is a first-order differential operator with differentiable coefficients containing derivatives only in the directions tangent to \(\Gamma\). One formulates a similar problem for harmonic functions regular in the exterior of \(D\). The conditions for the solvability and the number of solutions to these problems are treated here and the following particular cases are considered as illustrations: Solve the oblique derivative problem (1) if \(D\) is the unit ball \(x^2+ y^2+ z^2< 1\), with the boundary conditions \(u_2=b\) or \(xu_x+ yu_y+ zy_z= b\), and (2) in an axially symmetric domain \(D\) with the boundary conditions \(2z (xu_x+ yu_y+ zu_z)- (x^2+ y^2+ z^2) u_z+ zu= b\) or \(xu_x+ yu_y+ (z-z_0) u_z= b\), where the solution depends on whether \((0,0,z_0)\) is inside or outside \(D\).
reviewed by: Victor Anandam
Mathematics Subject Classification ID: 31B20
Mathematics Subject Classification ID: 35J25
zbMATH DE Number: 1465959
zbMATH Keywords: solvability conditions
zbMATH Keywords: oblique derivative problem
zbMATH Keywords: harmonic function
author: A. I. Yanushauskas
MaRDI profile type: Publication
cites work: Boundary-value problems for second-order equations and for systems of first-order equations
cites work: Q5596448
cites work: Q5332394
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full work available at URL: https://doi.org/10.1007/bf02465817
OpenAlex ID: W2057838812
description: scientific article; zbMATH DE number 1465959