Radial limits of solutions to elliptic partial differential equations

TL;DR

This paper investigates the boundary behavior of solutions to elliptic PDEs in starlike domains using harmonic line bundles and approximation theorems, revealing new insights into solution limits near boundaries.

Contribution

It introduces harmonic line bundles and approximation theorems to analyze elliptic PDE solutions' boundary limits in specific domains, advancing understanding of their asymptotic behavior.

Findings

Characterization of solution limits along radii in starlike domains

Development of harmonic line bundles as tools for analysis

Examples of approximation by elliptic PDE solutions in various domains

Abstract

For certain elliptic differential operators $L,$ we study the behaviour of solutions to $Lu=0,$ as we tend to the boundary along radii in strictly starlike domains in $\R^n, n\ge 3.$ Analogous results are obtained in other special domains. Our approach involves introducing harmonic line bundles as instances of Brelot harmonic spaces and approximating continuous functions by harmonic functions on appropriate subsets. These approximation theorems on harmonic spaces yield interesting examples for approximation by solutions of $Lu=0$ on some domains in $\R^n.$