6 problems

Let $U$ be a bounded open subset of $\mathbb C$, let $x\in U$, let $t=0$, and let $\phi$ be an admissible function. Let (a) denote the full-area-density condition and (c) the conve…

Let $U$ be a bounded open subset of $\mathbb C$ with $x\in U$, let $Y$ be the closure of $U$, let $t$ be a non-negative integer, and let $\phi$ be an admissible function. Denote by…

Propagation-of-smallness conjecture. The propagation of smallness for $|\nabla u|$ should hold from sets of positive $(d-2+\delta)$-Hausdorff content for every $\delta>0$.

Let $X$ be a compact, or even proper, metric space. Say that $X$ has locally small $n$-Hausdorff content when every point has a neighborhood with sufficiently small $n$-dimensional…

Let $c(B,m)$ denote the optimal constant in the codimension-one isoperimetric inequality, and let $C(B,m)$ denote the constant in the filling-radius bound involving…

Let $M^n$ be a closed connected topological submanifold of an $(n+1)$-dimensional Banach space $B^{n+1}$, and let $\Omega$ be the closure of the bounded connected component of…