Differential and Integral Equations

Higher order Neumann problems for Laplace's equation in two dimensions

Michael Renardy

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We discuss the boundary value problem $\Delta u=f$, $\partial^ku/\partial n^k=g$ in a bounded two-dimensional domain. For a smooth simply connected region, we prove that the only solutions of the homogeneous problem are harmonic polynomials of degree $k-1$. For multiply connected domains, this is true ``generically." In domains with corners, on the other hand, there are additional solutions.

Article information

Differential Integral Equations, Volume 15, Number 10 (2002), 1273-1279.

First available in Project Euclid: 21 December 2012

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J05: Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation [See also 31Axx, 31Bxx]
Secondary: 35J25: Boundary value problems for second-order elliptic equations


Renardy, Michael. Higher order Neumann problems for Laplace's equation in two dimensions. Differential Integral Equations 15 (2002), no. 10, 1273--1279. https://projecteuclid.org/euclid.die/1356060755

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