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Application of the Schwarz function to boundary problems for Laplace's equation
Author(s) -
Millar R. F.
Publication year - 1988
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.1670100106
Subject(s) - mathematics , mathematical analysis , analytic function , cauchy's integral formula , laplace's equation , laplace transform , integral equation , complex plane , bounded function , analytic continuation , domain (mathematical analysis) , function (biology) , simple (philosophy) , boundary value problem , cauchy problem , initial value problem , philosophy , epistemology , evolutionary biology , biology
Abstract A method is described for studying analytic boundary problems for the Laplace equation in a simplyconnected domain D of the plane. The aim is ultimately to obtain solutions in closed form when D is bounded by a simple analytic curve C with Schwarz function G . The basis for the procedure is that a functional F of the incompletely known boundary Cauchy data determines the solution and is analytic in D , and that there exists a further integral relation between these data and the analytic function G . Elimination of the unknown data from the latter relation by using the former leads to an integral on C that relates F to prescribed data and which, in some cases, may be solved for F in D . Some necessary properties of G are derived; in particular, it is shown that C must be a circle if G is analytic in D except for a single, simple pole (Theorem 1), or if a certain integral that appears prominently in this work is analytic throughout D (Theorem 2). The Dirichlet problem is treated in some detail. If C is a circle, a representation for the solution is obtained that is equivalent to one given by P. J Davis, and to the Poisson integral. When G is meromorphic in D and C is not a circle, Theorem 2 implies that the Davis result is not applicable, and the problem is reduced to that of solving Schröder's functional equation. Integral equations for F are obtained when C is an ellipse and G is multivalued in D . Other linear boundary value problems that can be handled are noted. A class of non‐linear Riemann‐Hilbert problems is described and studied briefly. For all these problems, closed solutions have been obtained when C is a circle; it is not clear whether closed solutions for other forms of C can be found without the explicit use of conformal mapping techniques. Finally, some possible generalizations to different equations and more complicated geometries are mentioned.