7 problems

Let $\Omega\subset\mathbb{R}^d$ be a bounded domain with real-analytic boundary, let $K\subset\Omega$ be compact, and let $U_k^{(0)}$ denote bulk eigenfunctions of the Dirichlet-to…

Let a curvilinear polygon have $n$ vertices with interior angles $\alpha_1,\dots,\alpha_n$. For $\alpha\in(0,2\pi)$, define … and … Form the multiset…

Let $\Omega\subset\mathbb{R}^d$ be a bounded Lipschitz domain, and let $\sigma^{(\Lambda)}$, $\Lambda\le 0$, be a fixed analytic eigenvalue branch of the Dirichlet-to-Neumann opera…

Let $\Omega\subset\mathbb{R}^d$ with $d=2$ or $d=3$ be a bounded, smooth, connected, and simply connected domain. Let $\rho_i>0$ be density fields and $c_i$ be generic stiffness te…

Removable-singularity conjecture. The roots of $\Omega+n\omega_0$ that must be removable singularities of $\widehat U_n(k)$ are precisely the two roots lying along $\partial D^-$,…

Let $\phi: \mathbb{R}^{n-1} \times \mathbb{R}^+ \to \mathbb{R}$. Suppose that $\phi(s,0) \equiv 0$. Let $f: \mathbb{R}^{n-1} \to \mathbb{R}$, and suppose … Here $g$ is the one-dime…

Let $(M,g)$ be an Einstein manifold with boundary, and let the Dirichlet-to-Neumann (DN) map associate boundary data for the Laplace–Beltrami equation to the corresponding normal d…