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. (Lean)

Manuscript

Strict hot spots and absence of interior critical points on smooth simply connected planar domains

September 24, 2026 · source · cite as \cite{OAI:Strict-hot-spots-and-absence-of-interior-critical-points-on-smooth-simply-connected-planar-domains-September-24-2026}

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.

Result 369 · Partial differential equations. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.