12 problems

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 $\Omega$ be an exterior sound-soft Helmholtz domain with boundary $\Gamma=\partial\Omega$, and let $u$ be the scattered-field solution represented by the method of fundamental…

The technical-loss conjecture. For every $k_0>0$ and $s\in\mathbb R$, there is $C>0$ such that, for all $N>0$, $k>k_0$, and $0<\epsilon<1$,

Assume $k_\epsilon>1/\sqrt{2}$ and define … This number is a pole of the Drude–Lorentz term, and the transmission eigenvalues are discrete outside the compact region $K$ described…

Uniform stable approximation conjecture. The sequence $\boldsymbol{\Psi}_{P}$ is a stable approximation for $\mathcal{A}_{P}$: there exist $s=s(P)\geq 0$ and $C=C(P)\geq 0$ such th…

Let $\sigma_T(\kappa,H,ell)>0$ be the parameter satisfying … Here $\sigma_T$ measures the quasi-orthogonality of $g$ on $Y$, and depends on the wavenumber through the wavenumber-depen…

Fundamental-solution conjecture. The candidate function $\mathsf S_{R,\beta}^{d,-}(\mathbf x,\mathbf x^\prime)$ is a fundamental solution for the Helmholtz operator…

Let $\Omega y$ be a bounded Lipschitz domain in $R^d$, let $a=1$, and let $cin L^{\infty}(\Omega)$ satisfy … For $omegageq \omega00$, consider the heterogeneous Helmholtz solution…

Let the obstacles be planar scatterers, and let the scattered field be defined in the exterior of the obstacles and on their boundaries. Assume that the scattered field is analytic…

Let $\Omega$ be the domain of the Helmholtz boundary-value problem, and let $\Gamma$ be any Jordan curve enclosing $\overline{\Omega}$ with positive distance from $partialOmega$. Cho…

Let $\Omega$ be a simply connected domain with analytic boundary, and let $GammaR={z:rhoz=R}$ be the curve of points with conformal radius $R1$. Choose the MFS charge points equal…

Let $\Omega$ be a simply connected domain with analytic boundary, and let $rhoz$ denote the conformal radius of $z$ with respect to $\Omega$. Place the MFS points equally spaced in…