The hot spots conjecture on Gaussian spaces

We study the hot spots conjecture for domains in the Gaussian space $({\mathbb{R}}^n,(2\pi {)}^{-n\mathrm{/}2}{e}^{-\mathrm{\mid }x{\mathrm{\mid }}^2\mathrm{/}2}dx)$ for $n\ge 2$. Given a bounded domain $\mathrm{\Omega }$ with a piecewise smooth boundary, we consider the first nontrivial eigenfunction of the Ornstein–Uhlenbeck operator $L_{\gamma }=\mathrm{\Delta }-⟨x,\mathrm{\nabla }⟩$ subject to Neumann or mixed Dirichlet–Neumann boundary conditions, and prove that its extrema are attained only on the boundary $\mathrm{\partial }\mathrm{\Omega }$. More precisely, we establish the conjecture for two classes of domains: (i) lip domains in Gaussian spaces with mixed boundary conditions, and (ii) $n$-symmetric domains whose intersection with some orthant is a lip domain. As a corollary, we show that any first nontrivial Neumann eigenfunction of a $2$-symmetric domain in the two-dimensional Gaussian space has no interior extrema, provided the second Neumann eigenvalue is simple. Our approach is based on a variational principle for the Hodge Laplacian on weighted manifolds and the Hodge decomposition of differential $1$-forms on Lipschitz domains, extending the variational method of Kennedy–Rohleder from the Euclidean setting to Gaussian spaces. Although de Dios Pont has shown that the hot spots conjecture can fail for certain convex domains endowed with suitable log-concave measures, our results identify broad classes of domains for which the conjecture remains valid in Gaussian spaces.