Bergman and Bauer Operators for Elliptic Equations in Two Independent Variables

Abstract

Object of this talk is to discuss some new results in solving the linear second order elliptic partial differential equation given in the canonical form

$$ \Delta u + \alpha \left( {x,y} \right){u_{{x + }}}\beta \left( {x,y} \right){u_y} + \gamma \left( {x,y} \right)u = 0 $$

with real analytic coefficients on a simply connected domain. First the equation is transformed in a suitable complex form, the formally hyperbolic differential equation

$$ {\upsilon_{{z\varsigma }}} + {a_1}\left( {z,\varsigma } \right){\upsilon_z} + {a_2}\left( {z,\varsigma } \right){\upsilon_{\varsigma }} + {a_3}\left( {z,\varsigma } \right)\upsilon = 0 $$

, in which it can be solved by different integral operators and the differential operators of K. W. Bauer. In this paper the integral operators of St. Bergman are discussed. M. Kracht and E. Kreyszig could prove that Bauer’s operators, which possess quite another structure than these of Bergman, are a special case of Bergman’s operator with polynomial kernels (Lemma 3). Such differential and integral operators for representing solutions of our equation provide a way of applying function theoretic methods and results to the study of various general properties of solutions. The rate of success of this approach depends on the construction of suitable kernels.

In this paper the kernels of the first kind of the Bergman operators are calculated for two differential equations which are adjoint to each other. By using Lemma 3 the differential operator solutions are obtained.