Asymptotic first boundary value problem for elliptic operators

TL;DR

This paper explores boundary value problems for solutions of elliptic equations, extending classical results for holomorphic functions to more general elliptic operators on Riemann surfaces and manifolds.

Contribution

It generalizes the boundary value problem framework from holomorphic functions to solutions of elliptic equations on complex and Riemannian geometries.

Findings

Established boundary value problem solutions for elliptic operators.

Extended classical boundary behavior results to Riemann surfaces.

Provided new insights into elliptic PDE boundary conditions.

Abstract

In 1955, Lehto showed that, for every measurable function $\psi$ on the unit circle $\mathbb T,$ there is a function $f$ holomorphic in the unit disc, having $\psi$ as radial limit a.e. on $\mathbb T.$ We consider an analogous problem for solutions $f$ of homogenous elliptic equations $Pf=0$ and, in particular, for holomorphic functions on Riemann surfaces and harmonic functions on Riemannian manifolds.