Singular Elliptic Equations Research Articles

Cauchy's problem for a singular parabolic partial differential equation

In recent years analytic function theory has been shown to play a basic role in the investigation of existence and uniqueness theorems for solutions to elliptic partial differential equations ([6], [8], [17]). An approach which has proved particularly fruitful is that of integral operators ([l], [12], [17]), from whose use complete families of solutions can be obtained, thus enabling one to construct the Bergman kernel function and solve the Dirichlet and Neumann problems ([2]). For singular elliptic equations serious difficulties arise due to the nonregularities in the kernels of such operators, as well as the failure of Green’s representation to hold in a neighborhood of the singular curve. In such cases recourse is often made to the use of operators whose path of integration is a contour in the complex plane ([7], [15]). In this paper we apply integral operator techniques in conjunction with function theoretic methods to establish an existence, uniqueness, and representation theorem to Cauchy’s problem for the singular parabolic equation

A contribution to the analytic theory of partial differential equations

The classical theory of functions of a complex variable plays a central role in the study of ordinary differential equations, and such topics as singularities of the regular type and equations of the Fuchsian class are brilliant examples of the power and beauty of this union. What has been only partly realized up to now is that complex variable theory is no less an indespensible tool in the investigation of the basic questions of existence and uniqueness of solutions for boundary value problems in partial differential equations [IO]. As in the situation with ordinary differential equations the necessity of examining solutions in the complex domain is not completely apparent until differential equations with singular coefficients are considered. It is the purpose of this paper to consider as an example a particular singular elliptic partial differential equation related to Bessel’s equation and to show that in order to obtain a uniqueness theorem for the exterior Dirichlet problem it is necessary to impose conditions on the solution in the complex plane.

Fundamental solutions for a class of equations with several singular coefficients

This paper is concerned with a class of singular elliptic partial differential equations related to the operator of Weinstein's generalized axially symmetric potential theory (GASPT) [1, 2] which has numerous applications.

Analytic Theory of Singular Elliptic Partial Differential Equations

are defined in D. In this connection, D, is a generic symbol for any of the partial derivatives of order t,. It will be assumed that f C L2 in D. Throughout this work any statement to the effect that a complex valued function q c L2 in a. domain R will be in the sense that q is measurable and that lq 12 is integrable over R. More precise hypotheses will be introduced in the sequel (see hypothesis (2.13)). The type of problems, relating to (1.1), which we shall study are suggested, in part, by T. Carleman's' investigation of a Schrddinger wave equation, which in the sequel will be referred to as (C). Carleman utilizes for this purpose the methods of his theory of singular integral equations2 in the sequel referred to as (C1). The case treated in (C) is