Dirichlet's problem with entire data posed on an ellipsoidal cylinder

TL;DR

This paper studies the Dirichlet problem for harmonic functions in an ellipsoidal cylinder with entire data, showing solutions extend as entire harmonic functions under certain growth conditions.

Contribution

It establishes conditions under which the Dirichlet problem with entire data admits solutions that are entire harmonic functions, extending previous results to ellipsoidal cylinders.

Findings

Solutions exist as entire harmonic functions when data has order less than one.

The work extends the theory of harmonic extension in ellipsoidal geometries.

Provides new insights into boundary value problems with entire data.

Abstract

We consider the Dirichlet problem in an ellipsoidal cylinder when the data function is entire. Under an additional assumption that the order of the data function is less than one, we show that there is a solution that extends as an entire harmonic function.