7 problems

Let $\Omega\subset\mathbb{R}^3$ be a compact domain, and let $\epsilon\in C^2(\mathcal{S}^2,\Omega)$ be an elastic strain field in an isotropic medium that satisfies equilibrium an…

Let $\Omega$ be an axially symmetric planar convex domain centered at the origin, and let the fail points be the points where the stress $|\nabla u|$ is maximal, with $u$ the torsi…

Let $\Omega$ be a bounded convex domain, and let the load $f \in \mathcal{M}(\overline{\Omega};\mathbb{R})$ be finitely supported. An optimal string system consists of a pair…

Let $E(3)$ be the space of three-dimensional stiffness tensors, let $F$ map a stiffness tensor to its slowness polynomial, and write the target as $\mathbb{R}[p_1,p_2,p_3]$. The sl…

Homeomorphism conjecture. Every minimizer $h_\circ\in\mathscr M^p(\overline{\mathbb{X}},\overline{\mathbb{Y}})$ is a homeomorphism.

Let $r\in\mathbb{Q}_{>1}\setminus\mathbb{N}$, and write $r=\mathsf{n}(r)/\mathsf{d}(r)$ in lowest terms. Let $S_r$ be the cyclic rational semiring generated by the geometric sequen…

Let $\Omega\subset\mathbb{R}^3$ be a Lipschitz domain, let $\emptyset\neq\Gamma\subset\partial\Omega$ be relatively open, and let…