S. L. Ma'u - Department of Mathematics, University of Auckland, Private Bag 92019 Auckland, New Zealand (email)
Abstract: 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: Helmholtz, spherical means, averaging method, boundary value problem.
Published: April 2003.