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Examines critical points of the Neumann Laplacian eigenfunction in convex domains, indicating spatial constraints for hot spots.
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.
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Jonathan Rohleder (2026) studied this question.
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