Construction and use of reproducing kernels for boundary and eigenvalue problems in the plane using pseudoanalytic function theory

We show how the Bergman-type reproducing kernels for the elliptic operator D = div pgrad+ q with variable coefficients defined in a bounded domain in the plane can be constructed using pseudoanalytic function theory and in particular pseudoanalytic formal powers. Under certain conditions on the coefficients p and q and with the aid of pseudoanalytic function theory a complete system of null solutions of the operator can be obtained following a simple algorithm consisting in recursive integration. Then the complete system of solutions is used for constructing the corresponding reproducing kernel. We study theoretical and numerical aspects of the method.