On an Iterative Scheme for the Spinorial Yamabe Equation on Manifolds with Boundary

2025/06/23 by Trébuchon, Eric
#53C27 #58F30 #58J32 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2506.18546

Abstract

We study the existence of solutions to the spinorial Yamabe equation -- that is, the Euler--Lagrange equation associated with the conformal invariant introduced by S. Raulot -- for compact manifolds with boundary. For the inhomogeneous equation, we employ an iterative scheme to establish existence under smallness assumptions on the relevant parameters. Using bootstrapping methods, we extend the regularity of the solution to C^∞ away from its zero set in the interior, and up to the boundary in the case of Shapiro--Lopatinski boundary conditions.

Citations

• Local Boundary Conditions for Dirac-type operators
• Solving the Yamabe Problem by an Iterative Method on a Small Riemannian Domain
• Hitchhiker's guide to the fractional Sobolev spaces
• Hitchhikerʼs guide to the fractional Sobolev spaces
• An existence theorem for the Yamabe problem on manifolds with boundary
• Nonlinear Dirac equations on Riemann surfaces
• Green functions for the Dirac operator under local boundary conditions and applications
• Dirac-Harmonic Maps
• Dirac-harmonic maps

Related

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• A variational analysis of the spinorial Yamabe equation on product manifolds
• Non-compactness results for the spinorial Yamabe-type problems with non-smooth geometric data
• Spinorial Yamabe-type equations and the Bär-Hijazi-Lott invariant
• Constructions of Delaunay-type solutions for the spinorial Yamabe equation on spheres