104 problems

Let $\O\subset\mathbb R^2$ be a bounded domain, and let $u$ be a second eigenfunction of the Dirichlet Laplacian eigenvalue problem on $\O$. A closed nodal line is a closed compone…

Let $(\Omega,g)$ be an $n$-dimensional compact $C^{\infty}$ Riemannian manifold with boundary $M$. Let ${\mathcal D}$ be the Dirichlet-to-Neumann operator on $M$, and let $u_h$ be…

Let $u$ be a harmonic function on the unit ball in $\mathbb{R}^{n}$ such that $u(0)=0$, and let $\mathcal{H}^{n-1}$ denote the Hausdorff measure of its nodal set. Nadirashvili's co…

Let $(M,g)$ be a compact $C^{\infty}$ Riemannian manifold of dimension $n$, let $\varphi_{\lambda}$ be an $L^2$-normalized eigenfunction satisfying … and let…

Let $M$ be a compact $n$-dimensional Riemannian manifold, let $u_\lambda$ satisfy … where $\lambda$ is the eigenvalue, and write $Z(u_\lambda)=\{u_\lambda=0\}$ for its zero set. Ya…

Let $(\mathcal{M},g)$ be a manifold with a smooth metric, let $E_n$ be a Laplace eigenvalue, let $g_n$ be an associated eigenfunction, and let $\mathcal{Z}(g_n)$ denote the volume…

Yau's conjecture. The Hausdorff measure of the nodal set of $u$ satisfies

Let $E$ be an eigenvalue of the Laplacian on the $d$-dimensional torus, and let $\mathcal{N}$ be the dimension of the eigenspace corresponding to $E$. Consider arithmetic random wa…

Let $(M,g)$ be a compact Riemannian surface without boundary, let $\Delta_g$ be its Laplace–Beltrami operator, and let $f_{\lambda}$ be a Laplace eigenfunction with eigenvalue…

Let $(M,g)$ be a smooth boundaryless compact connected Riemannian manifold of dimension $d$, let $\psi_\lambda$ be an eigenfunction with eigenvalue parameter $\lambda$, and let…

Let $M$ be a closed compact Riemannian manifold, and let $u_\lambda$ be a $\lambda$-eigenfunction on $M$. Denote its nodal set by $\{u_\lambda=0\}$ and its $(n-1)$-dimensional volu…

Let $u$ solve the divergence-form elliptic equation … in $B(0,1)$, with … and … Let $Z(u)$ be the nodal set, $S(u)$ the singular set, and $\beta_u$ the frequency quantity used in t…

Let $\cal H_\ell$ be the eigenspace of spherical harmonics of degree $\ell$ on the sphere, and let $\mu(u)$ denote the number of nodal domains of a spherical harmonic $u$. Leydold'…

Let $M$ be a smooth compact $n$-dimensional manifold, let $\Delta_M$ be its Laplace operator, and let $u_\lambda$ be a corresponding eigenfunction satisfying … Write…

Genericity conjecture. Triple intersection points are conjectured to be generic for large $k$ in spectral partitions.

Let $(\mathcal{M},g)$ be an $n$-dimensional compact smooth Riemannian manifold, let $\varphi_\lambda$ be a Laplace eigenfunction with eigenvalue $\lambda>0$, and let … be its nodal…

Yau's conjecture. There are constants $c,C>0$ such that

Let $(M,g)$ be a compact $C^{\infty}$ Riemannian manifold without boundary, and let $u$ be an eigenfunction of the Laplace–Beltrami operator with eigenvalue $\lambda$. Its nodal se…

Yau's conjecture. There are positive constants $c_1,c_2$, depending only on $(M,g)$, such that

Let $(M,g)$ be a compact Riemannian manifold of dimension $d$, and let $e_\lambda$ be a real eigenfunction with eigenvalue parameter $\lambda$. Define its nodal set by … and write…

Let $M$ be a closed Riemannian manifold of dimension $n$, let $\lambda$ be an eigenvalue of the Laplace operator, and let $H_{n-1}(\lambda)$ denote the $(n-1)$-dimensional Hausdorf…

Let $(M,g)$ be a real analytic Riemannian manifold with ergodic geodesic flow, and let $\{\varphi_j\}$ be a density-one sequence of ergodic eigenfunctions. For a test function $f$,…

Let $D\subset\mathbb{R}^2$ be a convex domain symmetric with respect to both coordinate axes. Let $\lambda_n$ and $\varphi_n$ be the eigenvalues and eigenfunctions of the Cauchy pr…

Nodal-set dimension conjecture. The nodal set of $\varphi$ has Hausdorff dimension at most $n-2$.

Let $\Omega\subseteq\mathbb{R}^2$ be a connected planar domain, and let $u_3$ be an eigenfunction corresponding to a simple third eigenvalue $\lambda_3$ of the Dirichlet Laplacian…