An averaging method for the Helmholtz equation

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2003, 2003(Special): 604-609. doi: 10.3934/proc.2003.2003.604

An averaging method for the Helmholtz equation

1.	Department of Mathematics, University of Auckland, Private Bag 92019 Auckland, New Zealand, New Zealand

Published April 2003

Abstract
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The well-known J. Schauder result on the existence of Lip$_\alpha(bar(\Omega))$ solutions of the Dirichlet problem for bounded domains with smooth boundaries is true for the Helmholtz equation $\Delta u + \lambda u = 0$ for $\lambda =< 0$. We suggest a method of constructing the solution based on an averaging procedure and mean-value theorem. We show some conditions under which, for $0 < \alpha < 1$, and $\lambda =< 0$, a sequence of iterated averages of an initial approximation converges geometrically to the solution.

Keywords: averaging method, spherical means, Helmholtz, boundary value problem..

Mathematics Subject Classification: Primary: 35J05, 35A35; Secondary: 65N0.

Citation: S. L. Ma'u, P. Ramankutty. An averaging method for the Helmholtz equation. Conference Publications, 2003, 2003 (Special) : 604-609. doi: 10.3934/proc.2003.2003.604

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