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An averaging method for the Helmholtz equation

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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.
    Mathematics Subject Classification: Primary: 35J05, 35A35; Secondary: 65N06.


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