16 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…

Let $D$ be a convex domain in ${\bf R}^n$, with $n\geq 1$, and let a second Neumann eigenfunction be an eigenfunction corresponding to the second eigenvalue of the Laplacian on $D$…

Suppose that $D$ is a triangle with vertices $C_1,C_2,C_3$, and that the angle at $C_3$ is greater than $\frac{pi}{2}$. Let $C_{12} in \overline{C_1C_2}$ be the unique point such t…

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

Let $\Omega_\epsilon$ be the perturbed symmetric dumbbell domain, with $M$ the length parameter and $g$ the neck-defining function. Let $\{\mu_k\}_{k=1}^\infty$ denote the Neumann…

Jerison's monotonicity conjecture. The monotonicity property holds for centrally symmetric convex domains. The source notes that eigenfunctions can fail to be monotone in general.…

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…

Let $D$ be a Lipschitz planar domain in $\mathbb{R}^d$, and let $\varphi_{2}$ be a second Neumann eigenfunction on $D$. Hot-Spots conjecture. The maximum and minimum of…

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…

Let $D$ be a domain and let $u$ be a Neumann eigenfunction on $D$, meaning that $u$ satisfies the Neumann eigenvalue problem on $D$. Schiffer's conjecture. If $u$ is constant on th…

Let $D$ be a simply connected domain, and let $u_2$ be a second eigenfunction of the Neumann eigenvalue problem on $D$. Simply connected-domain conjecture. No simply connected doma…

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…

Let $\Phi_{p,q}^\theta$ be the eigenfunctions of the Neumann problem in the square, and suppose the given eigenspace has dimension $2$. Five-angle conjecture. The maximal number of…

Let $M$ be a compact Riemannian manifold with boundary, and let $u$ be a Neumann eigenfunction with eigenvalue $\lambda^2$. For a fixed constant $C$, consider every geodesic ball i…