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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Removing Boundary Smoothness from the Hot-Spots Conclusion in the Simple-Eigenvalue CasearXiv.orgKrzysztof BurdzyOct 8, 2026in the index
We give a reduction from a smooth-domain theorem excluding interior critical points of first positive Neumann eigenfunctions to a non-strict hot-spots conclusion on arbitrary bounded simply connected planar domains with compact Neumann resolvent and simple first positive eigenvalue. The main…