An operator-theoretic existence proof of solutions to planar Dirichl\'et problems

TL;DR

This paper presents an operator-theoretic approach to prove the existence of solutions for planar Dirichlet problems, providing an iterative method with explicit error bounds and a closed-form solution without boundary smoothness assumptions.

Contribution

It introduces a constructive, operator-theoretic proof for Dirichlet problems in planar Jordan domains with multiple boundary curves, including an iterative solution method and explicit error estimates.

Findings

Constructive proof of solution existence using operator theory

Iterative method with explicit error bounds

Closed-form solution for Dirichlet problems

Abstract

By using some elementary techniques from operator theory, we prove constructively prove the existence of solutions to Dirichl\'et problems for planar Jordan domains with at least two boundary curves. An iterative method is thus obtained, and explicit bounds on the error in the resulting approximations are given. Finally, a closed form for the solution is given. No amount of differentiability of the boundary is assumed.