The hot spots conjecture for smooth simply connected planar domains

Theorem 1.1: on every bounded simply connected planar domain with $C^\infty$ boundary, every nonzero eigenfunction for the first positive Neumann eigenvalue has nonvanishing gradient in the interior, so its global extrema lie only on the boundary; this holds even when the eigenvalue is multiple and with no convexity or symmetry assumption. It does not treat nonsmooth (for example Lipschitz or polygonal) simply connected domains, domains with holes, or higher dimensions.

Theorem 1.1: on every bounded simply connected planar domain with $C^\infty$ boundary, every nonzero eigenfunction for the first positive Neumann eigenvalue has nonvanishing gradient in the interior, so its global extrema lie only on the boundary; this holds even when the eigenvalue is multiple and with no convexity or symmetry assumption. It does not treat nonsmooth (for example Lipschitz or polygonal) simply connected domains, domains with holes, or higher dimensions.