6 problems

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 simpl…

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 counterexample…

Critical-point conjecture. If $T$ is not an equilateral triangle, then a second Neumann eigenfunction of $T$ has a critical point if and only if $T$ is an acute triangle that is no…

Let $D$ be a non-equilateral triangle, and let $u$ be a second Neumann eigenfunction on $D$. Call $D$ superequilateral if it is isosceles with aperture angle greater than…

Let $u$ be a Neumann eigenfunction on a Platonic solid. Boundary-sign conjecture. Neumann eigenfunctions that are nonnegative on the boundary of a tetrahedron or octahedron do not…

Consider the Neumann problem in the square and a sequence of eigenfunctions associated with an infinite sequence of eigenvalues. Nodal-domain divergence conjecture. The number of n…