Boundary expansion for the Loewner-Nirenberg problem in domains with conic singularities

TL;DR

This paper investigates the asymptotic behavior of solutions to the Loewner-Nirenberg problem in domains with conic singularities, providing detailed expansions and analyzing eigenvalues and eigenfunctions on singular spherical domains.

Contribution

It introduces new asymptotic expansion techniques for solutions in conic domains and studies eigenvalue growth and eigenfunction estimates on singular spherical surfaces.

Findings

Established asymptotic expansions in conic domains with singularities

Analyzed eigenvalue growth for elliptic operators on singular spherical domains

Provided estimates for eigenfunctions in domains with conic singularities

Abstract

We study asymptotic behaviors of solutions to the Loewner-Nirenberg problem in domains with conic singularities and establish asymptotic expansions with respect to two normal directions simultaneously. The spherical domains over which cones are formed are allowed to have singularities. An elliptic operator on such spherical domains with coefficients singular on boundary play an important role. Key step is the study of the eigenvalues growth and eigenfunctions estimates.