On hot-spots free subregions of convex domains

2025/06/27 by Rohleder, Jonathan
#Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2506.22184

Abstract

The second eigenfunction of the Neumann Laplacian on convex, planar domains is considered. Inspired by the famous hot spots conjecture and a related result of Steinerberger, we show that potential critical points of this eigenfunction (and, in particular, interior "hot spots") cannot be located "near the center" of the domain. The region in which critical points are excluded is described explicitly.

Related

• Hot Spots in Convex Domains are in the Tips (up to an Inradius)
• Critical points of second Neumann eigenfunctions on some convex domains in two-dimensional space forms
• The hot spots conjecture for some non-convex polygons
• A counterexample to the "hot spots" conjecture
• Convex sets can have interior hot spots