A note on hot-spots free subregions of convex domains

TL;DR

This paper investigates the location of critical points of the second Neumann Laplacian eigenfunction in convex planar domains, showing they cannot be near the domain's center, thus providing explicit exclusion regions.

Contribution

It provides a new explicit description of regions where critical points of the second eigenfunction cannot occur in convex domains.

Findings

Critical points are excluded from regions near the domain's center.

Explicit regions where hot spots cannot be located are described.

Supports and extends the hot spots conjecture in convex domains.

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.