7 problems

Let $\Omega\subset\mathbb{R}^2$ and let $u\in BV(\Omega;\mathbb{R}^m)$ belong to a suitable class of maps of bounded variation. For open sets $A\subset\Omega$, consider the strictl…

Let $f\in BV_{||0||}(\mathbb{R}^2)$, and let the limit referred to as … is locally uniform at every point of continuity of $f$. The conjecture concerns strengthening pointwise conv…

Let $(\mathsf{X},\mathsf{d},\mathfrak{m})$ be a $PI$ space, let $\Omega\subset\mathsf{X}$ be open, and let $u\in\mathsf{BV}(\Omega)$. Define the area of $u$ in a measurable set…

Let $\Omega\subset\mathbb{R}^d$ be an open set. The space $SBD^2(\Omega)$ consists of special functions of bounded deformation with square-integrable elastic strain, and…

Let $\Omega=\mathbb{R}^N$, let $u\in\mathrm{BV}^2(\mathbb{R}^N)$, and let $H_nu$ denote the non-local Hessian of $u$ associated with the approximation of the identity used in the p…

Asymptotic formula conjecture. For every $f\in SBV_{\rm loc}(\mathbb{R}^n)$,

$N$-term approximation conjecture. The approximation error measured in $L^2$ for $\mathit{f}\in BV$ should decay as