20 problems

Let $\Omega$ range over bounded Lipschitz domains of fixed volume, and let $R^\Omega_\alpha$ denote the Robin Laplacian on $\Omega$ with fixed parameter $\alpha<0$. Reverse Faber–K…

Let $\Omega \subset \mathbb{R}^2$ be a planar domain with $n$ corner points $y_1, \dots, y_n$ on its boundary $\Gamma$, and let $\alpha_j$, $j=1,\dots,n$, denote the inner half-ang…

Let $\beta(\gamma,d)$ be the critical Robin parameter from Theorem, and let $\beta_W(\gamma,d-1)$ be the point where the second term in the two-term Weyl formula changes sign. Stri…

Let $\beta(\gamma,d)$ be the critical Robin parameter from Theorem, and let $\beta_W(\gamma,d-1)$ be the value at which the second term in the two-term Weyl formula changes sign. W…

Let $\Omega\subset\mathbb R^d$ be a domain, let $\beta$ be a Robin boundary parameter, and let $\gamma$ be the Riesz-mean order. Theorem gives a two-term semiclassical asymptotic e…

Let $\beta>0$ be a Robin parameter and let $\beta_k$ denote a threshold depending on the eigenvalue index $k$. An optimizer is a domain attaining the minimum of the $k$-th Robin ei…

Exponent-two conjecture. The exponent $2$ should be the optimal exponent in a quantitative estimate in terms of the Fraenkel asymmetry, instead of the exponent $3$ in the displayed…

Quadratic-exponent optimality conjecture. The quadratic exponent is optimal in these quantitative inequalities of spectral type.

Let $u_0$ be an initial function for the iterative scheme defined by … , at least when $R_m$ has a unique minimizer. The conjecture concerns convergence of an iteration whose bound…

Ashbaugh–Kielty conjecture. The minimizing potential is linear but not constant.

For the Robin Laplacian with negative boundary parameter, consider domains with fixed boundary area; in two dimensions restrict to simply-connected domains, while in higher dimensi…

Robin maximization conjecture. The ball maximizes $\lambda(\Omega,2,\beta)$ among all smooth bounded domains $\Omega$ of fixed volume.

Let $\Omega\subset\mathbb{R}^d$ be a bounded Lipschitz domain, and let $\lambda_1(\Omega,\sigma,\alpha,\beta,\gamma)$ be the first eigenvalue of the Robin bilaplacian with paramete…

Let $\Omega\subset\mathbb{R}^d$ be the domain and let $\mathcal{A}[\beta,\gamma]$ denote the operator associated with the Robin bilaplacian for parameters $\beta,\gamma$, with…

Andrews–Clutterbuck–Hauer conjecture.

Let $\Omega\subset\mathbb{R}^N$ be a suitable domain and let the Robin ground state be the positive first eigenfunction of the Laplace operator on $\Omega$ with positive Robin para…

Small-parameter stability conjecture. There exists $h_0>0$ such that for $0<h\leq h_0$, the Courant-sharp cases for the Robin problem are the same, except the fifth one, as those f…

Let $0Omega$ be the unit disk, let $0betageq 0$, and let E^ be an optimal set for Problem … when $\Omega$ is the unit disk and $\beta\geq 0$. The statement is motivated by nume…

Robin eigenvalue ball-maximization conjecture. Among sets of a given measure, balls achieve the greatest eigenvalue $\lambda(\Omega,\alpha)$.

Let $\Omega\subset\mathbb{R}^\nu$ be a domain of fixed volume, and let $\Lambda(\Omega,2,\alpha)$ denote the quantity studied in the paper for the Robin problem, with parameter…