Hot spots conjecture for simply-connected planar and convex domains

Let $\Omega$ be a simply-connected planar domain, or a convex domain in $\mathbb{R}^n$, and let $u$ be a second Neumann eigenfunction of the Laplacian on $\Omega$.

Hot spots conjecture. The function $u$ attains its maximum only on the boundary $\partial\Omega$. The source presents this as the restricted formulation motivated by counterexamples to the original conjecture. The paper establishes the relevant triangle case, while the full assertion for all simply-connected planar domains and all convex domains in every dimension is not resolved in the supplied text.