4 problems

Rauch's hot spots conjecture. The extrema of solutions of the Neumann heat equation should tend toward $\partial\Omega$ as time tends to infinity; equivalently, $u$ should attain i…

Rauch's conjecture. The extrema of $u$ occur only on the boundary $\partial\Omega$.

For each dimension $d$, let $C_d$ be the hot spots ratio of $d$-dimensional convex sets: the supremum, over all convex bounded domains $\d7\subset\mathbb R^d$, of … Here $\c8_\d7$…

Let $(\Omega,D,N)$ be a triple where $\Omega$ is simply connected, $D$ is a Euclidean line segment, and … Say that $(\Omega,D,N)$ has no hot spots when each first mixed eigenfuncti…