Hot spots conjecture for convex planar domains

Determine whether, for every bounded convex planar domain Ω ⊂ R^2 with Neumann boundary conditions, any eigenfunction ψ2 associated with the first nontrivial Neumann eigenvalue μ2 of the Laplacian (−Δψ2 = μ2ψ2 in Ω, ∂ψ2/∂ν = 0 on ∂Ω) attains its global minimum and maximum only on the boundary ∂Ω.

Background

The hot spots conjecture (Rauch, 1970s) asserts that for the Neumann Laplacian on a bounded domain, the first nontrivial eigenfunction has no interior extrema and instead achieves its maximum and minimum on the boundary. In two dimensions, the conjecture has been proven for several specific classes of convex domains, including triangles and “long and thin” domains, and certain symmetric domains.

Despite these advances, the general case for convex planar domains remains unresolved. The present paper proves strong forms of the conjecture in higher dimensions for new classes (multidimensional lip domains and certain symmetric domains), but it does not resolve the classical open problem for all convex planar domains.