41 problems

Let $u^\varepsilon$ be the wave function and define its rescaling near the origin by … Assume that $w^\varepsilon$ is bounded in the appropriate weighted space and let…

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…

Numerical-error conjecture. The slight deviation of the relative errors from the expected $O(N^{-1})$ behavior at

Linear resolvent-growth conjecture. There exists a constant $C$ such that

Pollution in the $h$-FEM means that substantially more than $k^d$ degrees of freedom may be required to accurately approximate a scattering solution at frequency $k$. The boundary…

Let the fine and coarse discretizations in the two-grid finite-element method for Helmholtz problems have finite-element spaces, with the coarse finite-element space not contained…

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 $0<\epsilon<1$,

Baumann–Sauter conjecture. The Galerkin method applied to $A_k$ and $A'_k$ does not suffer from the pollution effect.

Chandler-Wilde–Spence–Galkowski–Smith's resolvent conjecture. There is a constant $C>0$ such that

Let $\rho(k)$ denote the stability constant for the Helmholtz problem at wavenumber $k$, and let $C$ and $k_0$ be positive constants. The best known general upper bound is…

Absence-of-trapped-modes conjecture. The absence of non-trivial square-integrable solutions, equivalently trapped modes, still holds for such junctions; the restriction to angles g…

Actual decay-rate conjecture. The actual decay rate is

Let $K$ be a compact set, let $\rho$ be as in Theorem wcont, and let $u$ be a measure supported by $K$ such that $calF{k}^{rho}(mu)neqemptyset$. Denote by $omega{k}^{rho}(mu)$ the…

Stable discrete approximation conjecture. The evaluation functional sequence $\{\Psi_{L,P}\}_{L\geq 0,\,P\in\mathbb{N}}$ is a stable discrete approximation for the space of Herglot…

Let $\boldsymbol{\Psi}_L$ be the sequence of approximation sets defined in the evaluation sets, and let $mathcal{A}L$ be the corresponding truncated approximation space. For a pre…

Mode-by-mode accumulation conjecture. For each $n=1,2,\ldots$, there exists $k_n\in\mathbb{C}\setminus\{\kappa\}$ such that

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…

Let $k_\epsilon>1/\sqrt{2}$, let … and let $K$ be the compact region described in the paper, with boundary symmetric about the imaginary axis. The complex transmission eigenvalues…

General-p phase-error conjecture. The coefficient and the corresponding phase difference satisfy

The authors' approach concerns active exterior cloaking for the Helmholtz equation with complex wavenumbers, including media with losses or gain. 3D cloaking conjecture. Their appr…

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…

Stable sampling conjecture. The associated sets of sampling functionals yield good approximation properties in $\mathcal{A}$; consequently, applying the Herglotz transform $T$, the…

Conjectured regular–singular decomposition. The regular part $u_R$ should admit the above decomposition into a non-oscillating $H^2$ component, an oscillating analytic component, a…

Let a sequence in the expansion of the four-subdomain product define a corresponding term in that product, and suppose the term is associated with a sequence of subdomains. Repeate…