The Cauchy problem and Hadamard's example in the ring

TL;DR

This paper constructs an integral representation for harmonic functions in a ring, proves the existence and uniqueness of solutions to the Cauchy problem for the Laplace equation in this domain, and provides an explicit integral solution.

Contribution

It introduces a new integral representation for harmonic functions in a ring and establishes existence and uniqueness results for the Cauchy problem in this setting.

Findings

Integral representation for harmonic functions in the ring

Existence and uniqueness of solutions to the Cauchy problem

Explicit integral solution for the Laplace equation in the ring

Abstract

Integral representation for harmonic function in the ring is constructed in this work. We prove the existence and uniqueness of solutions of the Cauchy problem for the Laplace equation in the ring. Integral representation for the solution of the Cauchy problem was found.