Hyperbolic Formulas in Elliptic Cauchy Problems

TL;DR

This paper develops explicit formulas for solving elliptic Cauchy problems using hyperbolic theory, extending classical methods to complex domains for the Laplace equation in cylindrical geometries.

Contribution

It introduces a novel approach by reducing elliptic problems to hyperbolic ones in complex domains, providing explicit solution formulas.

Findings

Explicit formulas for solutions to elliptic Cauchy problems

Extension of classical hyperbolic methods to complex domains

Application to Laplace equation in cylindrical geometries

Abstract

We study the Cauchy problem for the Laplace equation in a cylindrical domain with data on a part of it's boundary which is a cross-section of the cylinder. On reducing the problem to the Cauchy problem for the wave equation in a complex domain and using hyperbolic theory we obtain explicit formulas for the solution, thus developing the classical approach of Hans Lewy (1927).