Guess the Hot Spot: The hot spots conjecture for simply connected planar domains

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Guess the Hot Spot

based on Result #369: The hot spots conjecture for simply connected planar domains

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Difficulty:	🧠🧠🧠🧠🧠	Ages:	13 - ∞
Skills:	fluids, heat, waves	Levels:	1
Category:	Partial differential equations	Lean version:	YES! ✔
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The hot spots conjecture for simply connected planar domains. 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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LEVEL 1 Strict hot spots and absence of interior critical points on smooth simply connected planar domains

released 2026-09-24 | 2 theorems · 12 lemmas · 18 proofs · 7,656 words | PLAY LEVEL 1 » (pdf)

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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