10 problems

Let $\Omega\subset\mathbb{R}^d$ be any bounded set, and let the magnetic field be constant. Denote by $\lambda^{\rm DN}(A,\Omega)$ the ground-state energy of the magnetic Dirichlet…

Let $\Omega$ be a regular domain in $\mathbb{R}^2$, let $A_0$ be a magnetic potential whose magnetic field is constant and has norm $1$, and let $\lambda^{\rm DN}(bA_0,\Omega)$ den…

Let $\Omega$ be a regular domain in $\mathbb{R}^2$, and let $A_0$ be a magnetic potential whose magnetic field is constant and has norm $1$. Let $\Lambda_{bA_0}$ be the associated…

Let $D(0,1)\subset\mathbb{R}^2$ be the unit disk, let $\Lambda^{DN}(b)$ be the magnetic Dirichlet-to-Neumann map for a constant magnetic field of strength $2b$, and let…

Let $M$ be a Riemannian manifold with a boundary that need not be compact. The Dirichlet-to-Neumann operator and the associated mass-dependent quantities are as in Assumption … hol…

Let $\Omega$ be a polygonal domain, let $p$ be the parameter in the Dirichlet-to-Neumann operator, and let $V_k^{(p)}$ be a first Steklov eigenfunction with eigenvalue…

Let a triangle have angles $\alpha_0^{(0)}=\pi/12$, $\alpha_1^{(0)}=\pi/3$, and $\alpha_2^{(0)}=7\pi/12$, and let $c_0$ denote the first asymptotic coefficient for the Dirichlet-to…

Let a regular polygon have $N$ vertices and interior angle … For the asymptotic coefficients $c_k$ governing the first $N$ eigenvalues of the Dirichlet-to-Neumann operator, the reg…

Regularity conjecture. The regularity assumptions on $\Omega$ and $q$ required for this bound may be relaxed. The theorem is proved under smooth-boundary and smooth-potential assum…

Let $\Lambda$ be the Dirichlet-to-Neumann operator, let $P_\tau$ denote the spectral projection associated with the threshold $\tau$, and let $J$ be the operator mapping $v$ to…