Rauch's hot spot conjecture for convex and simply connected planar domains

Let $\Omega$ be a domain in $\mathbb{R}^n$, and let $u$ be a second Neumann eigenfunction on $\Omega$. Rauch's hot spot conjecture. If $\Omega$ is convex, or if $\Omega$ is a simply connected planar domain, then the maximum and minimum of $u$ are attained only on the boundary of $\Omega$. This is a positive case of the hot spot problem, which is false for general domains and has counterexamples among planar domains with holes.