8 problems

Let $\Omega$ be a ball, and let $u\in H_0(\operatorname{curl};\Omega)\cap L^p(\Omega,\mathbb{R}^3)$ be a ground state solution of … where $2<p<6$. Here, symmetry means the property…

Local embeddability conjecture. A CR 3-space admits locally a closed, non-zero section of its canonical bundle if and only if it is locally embeddable.

Fix $M>0$. Let $\eta\in(0,1)$ and $m\in\mathbb{Z}$. Let $J_2,J_3\geq J_1\geq J_{\mathrm{min}}\geq 1$, and let $C=C(m,M,J_1,J_2,J_3,\eta)>0$. For any $|a|\leq M$ and any solution…

Let ${\mathbb R}^3$ be divided into two domains $\Omega_1$ and $\Omega_2$ separated by a surface $\Gamma$, with real-valued permittivities $\epsilon_1>0$ and $\epsilon_2<0$. Set…

Let $\Omega$ be a convex domain. Let $c_{m,t}$ and $c_{m,n}$ denote the best constants in the tangential and normal Maxwell inequalities, respectively, and let $\lambda_1$ be the f…

Positive Sobolev regularity conjecture. For every bounded Lipschitz domain $Omega$, there exists $varepsilon>0$ for which the displayed inclusion holds. The paper states that this…

Let $\Gamma$ be the boundary of a strictly convex obstacle, let $\gamma(y)$ be the dissipation parameter on $\Gamma$, and suppose … Let $N(r)$ denote the counting function of negat…

Consider soliton solutions $u:\mathbb{R}^3\to\mathbb{R}^3$ of … The source proposes that such soliton solutions have cylindrical symmetry and are of the form specified by its equat…