Nonuniqueness for a fully nonlinear, degenerate elliptic boundary value problem in conformal geometry

2020/02/26 by Zhengyang Shan, Shan, Zhengyang
Mathematics · Computer Science · #Nonlinear Partial Differential Equations #Geometric Analysis and Curvature Flows #Advanced Mathematical Modeling in Engineering

paper · pdf · doi:10.48550/arxiv.2004.01514

Abstract

We study the problem of conformally deforming a manifold with boundary to have vanishing σ4-curvature in the interior and constant H4- curvature on the boundary. We prove that there are geometrically distinct solutions using bifurcation results proven by Case, Moreira and Wang. Surprisingly, our construction via products of a sphere and hyperbolic space only works for a finite set of dimensions.

Citations

• The Dirichlet problem for nonlinear second order elliptic equations, III: Functions of the eigenvalues of the Hessian

Related

• Nonuniqueness for a fully nonlinear boundary Yamabe-type problem via bifurcation theory
• Local bifurcation diagrams and degenerate solutions of Yamabe-type equations
• Bifurcation results for the Yamabe problem on Riemannian manifolds with boundary
• Classification theorems for solutions of higher order boundary conformally invariant problems, I
• A general nonuniqueness result for Yamabe-type problems for conformally variational Riemannian invariants