19 problems

Monotonicity conjecture. These clamped transmission eigenvalues are monotonically decreasing with respect to the measure of $D$.

Existence conjecture. Assume $\chi_1>0$. Then there exist $\delta,c>0$ such that every $\omega\in(0,\delta)$ is a generalized transmission eigenvalue with a corresponding eigenfunc…

Let $D$ be a polygonal domain that may have reentrant corners, and let Theorem 3.3 denote the theorem asserting the relevant limiting behavior of the first transmission eigenvalue…

Let $d=2$ or $d=3$, let $B_2$ be the ball of radius $2$, and let … A Born transmission-eigenvalue strip-finiteness conjecture. For any $C>0$, the strip … contains at most finitely…

Let $d=2$ or $d=3$, let $B_2$ be the ball of radius $2$, and let $m$ be the refractive-index function … A Born transmission-eigenvalue strip-finiteness conjecture. Any strip parall…

Let $d=2$ or $d=3$, let $k$ be the wave number, and let $k_\epsilon>0$ be the resonant frequency in the Drude–Lorentz term … A complex transmission-eigenvalue accumulation conjectu…

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…

Let $(\mega,V)$ be an inhomogeneous medium and let $\Gamma,\Gamma'\subset\partial\Omega$ be open boundary subsets in the partial-data transmission eigenvalue problem. For…

Let $\theta_V$ be the dataset of far-field patterns generated by a fixed incident direction $d\in\mathbb{S}^{n-1}$ and all positive frequencies and observation directions, as defin…

Circle minimization conjecture. The first and second interior transmission eigenvalue is minimized by a circle.

Let $\lambda_1,\lambda_2,\lambda_3,\ldots$ denote the interior transmission eigenvalues, ordered by … For an index of refraction $n>1$, the first absolute interior transmission eig…

Let $\Gamma$ be the curve appearing in Lemmas … , and suppose that $\Gamma$ contains a single non-analytical point. Single-point non-analyticity conjecture. Lemmas … remain valid u…

Let $v$ be a transmission eigenfunction on a domain with vertices and edges. Let $alpha(v,P)$ and $beta(v,E)$ denote the convergence orders at a vertex $P$ and an edge $E$, respect…

Let $u_0^{(i)}$ and $u^{(i)}$ be transmission eigenfunctions on a polygonal domain, and let a corner have interior angle $\theta$. Acute-corner vanishing conjecture. The vanishing…

Let $D$ be a domain, $n$ its refractive index, $v$ a transmission eigenfunction, and $P$ a singular point. Let $alpha(v,P)$ denote the order of convergence defined in the source. D…

Let $v$ be a transmission eigenfunction and let $P$ be a singular point at which the order of convergence $alpha(v,P)$ is defined. Convergence-order independence conjecture. The or…

Let $D$ be a domain with refractive index $n$, and let $u_0$ and $u$ be transmission eigenfunctions associated with a transmission eigenvalue. A point on the support of $1-n$ is ca…