The Nirenberg problem on half spheres: A bubbling off analysis

TL;DR

This paper conducts a detailed blow-up analysis of solutions to a Nirenberg-type problem on half spheres, revealing complex blow-up behaviors and connections to fluid dynamics and physics.

Contribution

It provides a precise characterization of blow-up points and rates, uncovering new phenomena such as non simple blow ups and their relation to vortex problems.

Findings

Identification of blow-up points and rates on half spheres.

Discovery of non simple blow up points influenced by vortex problems.

Unveiling connections between geometric analysis and fluid dynamics.

Abstract

In this paper we perform a refined blow up analysis of finite energy approximated solutions to a Nirenberg type problem on half spheres. The later consists of prescribing, under minimal boundary conditions, the scalar curvature to be a given function. In particular we give a precise location of blow up points and blow up rates. Such an analysis shows that the blow up picture of the Nirenberg problem on half spheres is far more complicated that in the case of closed spheres. Indeed besides the combination of interior and boundary blow ups, there are non simple blow up points for subcritical solutions having zero or nonzero weak limit. The formation of such non simple blow ups is governed by a vortex problem, unveiling an unexpected connection with Euler equations in fluid dynamic and mean fields type equations in mathematical physics.