19 problems

Sharp constant conjecture. The inequality holds with $C=1$ for every function $f:\mathbb{R}\to\mathbb{R}$ of bounded variation.

Liu–Pego's local convexity conjecture. Every such map $\phi$ must necessarily be locally convex in $\Omega$.

Let $M$ be the centered Hardy--Littlewood maximal operator on $fR$, and let $f\in\operatorname{BV}(fR)$ have limits $f(\infty)$ and $f(-\infty)$ at the two ends. Write…

Let $T_0,T_1$ be boundaryless integral $k$-currents. The boundaryless Whitney flat norm is … and the deformation distance is … Equality conjecture. Neglecting issues of domains, on…

Let $f\colon \mathbb{R}\to\mathbb{R}$ be a function of bounded variation, and define its centred Hardy–Littlewood maximal function by … Write … for the total variation. Sharp const…

Lipschitz-function conjecture. Every non-constant Lipschitz function is not of bounded variation with respect to definition $(A)$.

Let $F\in\mathcal{F}[a,b]$, let $A_F(x)$ denote the Fourier approximation limit associated with $F$, and let $F(x-0)$ and $F(x+0)$ be the left and right limits of $F$ at $x$. Write…

Let $\mathcal B[D]$ be a $\mathbb{C}$-elliptic operator and let $u$ be a function in $\mathrm{BV}^{\mathcal B}(\Omega)$. Write $S_u$ for the approximate discontinuity set and…

Let $\Omega$ be the domain, let $\mathscr{N}$ be a general target manifold with covering space $\mathscr{E}$, and let $\pi\colon\mathscr{E}\to\mathscr{N}$ be the covering map. For…

The paper considers homeomorphisms and several distributional versions of the adjugate of the gradient matrix, including the conditions used to characterize bounded-variation regul…

Nested-fractal variation conjectures.

Rectifiability conjecture. Property $\mathcal R$ holds in every equiregular Carnot–Carathéodory space of step $2$.

Characterization conjecture. The function $u$ belongs to $\mathcal E(Q)$ if and only if: (1) $u\in\operatorname{BV}(Q)$; (2) the outward unit normal $u$ is a function of bounded v…

Let $d>2$ and $N\geq 2$. For an isometric embedding $\Phi\colon{\mathbb{RP}^{d-1}}\to{\mathbb R}^D$, let $C^a(N,d)$ be the constant defined in Theorem 1.1 by the supremum involving…

Let $\Omega\subset\mathbb{R}^n$ be the domain, and let $SBD^p(\Omega)$ denote the space of special functions of bounded deformation with strain in $L^p$ and jump set of finite…

Let $u_0\in BV(\mathbb{R}^N)$ be initial data for the entropy solution $u$ of the flux-saturated problem considered in the paper. BV regularity conjecture. The condition…

Sharp variation conjecture. The constant in the inequality

Let $1<p<\infty$ and $1/p<\alpha<1$. For a nondecreasing sequence $\Lambda=\{\lambda_n\}_{n=1}^\infty$ of positive numbers, let $\Lambda BV$ denote the class of functions of bounde…

BV-structure conjecture. The bounded-variation structure of the velocity remains valid in the general case without a convexity assumption, or, more specifically, when the eigenvalu…