Complex Analysis of Real Functions V: The Dirichlet Problem on the Plane

TL;DR

This paper demonstrates that the correspondence between real functions on the unit circle and inner analytic functions can be used to prove the existence of solutions to the Dirichlet problem on the plane for a broad class of boundary functions, including discontinuous and unbounded ones.

Contribution

It introduces a general proof method for the Dirichlet problem using complex analysis, extending to non-integrable functions and various boundary shapes via conformal maps.

Findings

Existence of solutions for almost arbitrary integrable boundary functions.

Extension of existence results to non-integrable boundary functions.

Generalization to various boundary shapes through conformal transformations.

Abstract

In the context of the correspondence between real functions on the unit circle and inner analytic functions within the open unit disk, that was presented in previous papers, we show that the constructions used to establish that correspondence lead to very general proofs of existence of solutions of the Dirichlet problem on the plane. At first, this establishes the existence of solutions for almost arbitrary integrable real functions on the unit circle, including functions which are discontinuous and unbounded. The proof of existence is then generalized to a large class of non-integrable real functions on the unit circle. Further, the proof of existence is generalized to real functions on a large class of other boundaries on the plane, by means of conformal transformations.