Cherrier-Escobar problem for the elliptic Schroedinger-to-Neumann map

TL;DR

This paper investigates a geometric PDE problem related to the elliptic Schrödinger-to-Neumann map on 3D manifolds, establishing solvability conditions using topological and spectral assumptions.

Contribution

It introduces a new solvability result for the Cherrier-Escobar problem in the context of the Schrödinger-to-Neumann map, utilizing algebraic topological methods.

Findings

Solvability is achieved under specific spectral and positivity conditions.

The problem is formulated on compact 3D Riemannian manifolds with boundary.

The approach combines topological arguments with spectral analysis.

Abstract

In this paper, we study a Cherrier-Escobar problem for the extended problem corresponding to the elliptic Schroedinger-to-Neumann map on a compact 3-dimensional Riemannian manifold with boundary. Using the algebraic topological argument of Bahri-Coron, we show solvability under the assumption that the extended problem corresponding to the elliptic Schroedinger-to-Neumann map has a positive first eigenvalue, a positive Green function, and also verifies the strong maximum principle.