Pointwise bounds for the solutions of certain boundary-value problems

Abstract
The boundary-value problems considered are of the Dirichlet-Neumann type. The method now given for obtaining pointwise bounds of the solution and its derivatives is a compromise between the methods of Diaz and Greenberg on the one hand and Maple and Synge on the other; it appears to be simpler than either. The solution having been located on a hypercircle in function space, the pointwise bounds are obtained by taking the scalar product of the solution by certain vectors (Green’s vectors). Divergence of integrals due to the poles of the Green functions is avoided by the use of regular functions matching the Green functions on the boundary (Diaz-Greenberg device) instead of by cutting out spheres from the domain (Maple-Synge device).

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