%0 Journal Article
%T The Method of Fischer-Riesz Equations for Elliptic Boundary Value Problems
%A A. Alsaedy
%A N. Tarkhanov
%J Journal of Complex Analysis
%D 2013
%I Hindawi Publishing Corporation
%R 10.1155/2013/486934
%X We develop the method of Fischer-Riesz equations for general boundary value problems elliptic in the sense of Douglis-Nirenberg. To this end we reduce them to a boundary problem for a (possibly overdetermined) first-order system whose classical symbol has a left inverse. For such a problem there is a uniquely determined boundary value problem which is adjoint to the given one with respect to the Green formula. On using a well-elaborated theory of approximation by solutions of the adjoint problem, we find the Cauchy data of solutions of our problem. 1. Introduction The aim of this paper is to bring together two areas in which integral formulas like Green¡¯s formula for harmonic functions are of great importance. The first area is complex analysis where the method of integral representations was a central tool over the second half of the 20th century, see [1]. And the second area is the theory of elliptic boundary problems where the parametrix method led to the most refined results, such as local principle, -algebras of pseudodifferential boundary value problems [2], and so forth. The method of Fischer-Riesz equations can be specified within a larger approach which is usually referred to as the boundary element method. By this latter is meant a numerical method of solving boundary value problems which have been formulated as boundary integral equations. It can be applied in many areas of engineering and science including fluid mechanics, acoustics, electromagnetics, and fracture mechanics, see [3¨C6]. The boundary elements method attempts to use the given boundary conditions and other data of the problem to fit boundary values into the integral equation, rather than values throughout the space defined by a system of partial differential equations. Once this is done, the boundary integral equation can be used again to calculate numerically the solution directly at any desired point in the solution domain. More precisely, from the Cauchy data of the solution on the whole boundary one calculates readily the solution in the domain provided a left fundamental solution of the system is available in an explicit form, see for instance Lemma£¿£¿10.2.3 in [7]. The idea of the method of Fischer-Riesz equations goes back at least as far as [8]. The paper [9] was given by Picone as an invited address before the Second Austrian Mathematical Congress in Insbruck in 1949. He states in the introduction that he asked Fichera to write a certain part of the report. It is a crystallisation in the form of an abstract theory of some of the methods used by the authors and their
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