The hot spots conjecture for simply connected planar domains
OpenAI research collection · 1 manuscript · Source checked 2026-10-19
OpenAI's result summary
Proves a strict form of Burdzy's simply connected hot spots conjecture. On every smooth bounded simply connected planar domain, each nonzero eigenfunction for the first positive Neumann eigenvalue has no interior critical point, so all global extrema lie on the boundary. Eigenvalue multiplicity is allowed.
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MANUSCRIPT 01
Strict hot spots and absence of interior critical points on smooth simply connected planar domains
We prove the strict hot spots conjecture for smooth bounded simply connected planar domains. More precisely, every nonzero eigenfunction for the first positive Neumann eigenvalue has nonvanishing gradient in the interior, so all its global maxima and minima lie on the boundary. This holds even when the eigenvalue is multiple.
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