7 problems

Let $(M_t)_{t<T}$ be compact embedded hypersurfaces evolving by mean curvature, with $H>0$, and set $\tau=a-t$ for fixed $a\geq T$. Let $\Omega_t$ be the enclosed domains, let $u$…

Let $(M_t)_{t<T}$ be a family of compact embedded hypersurfaces in Euclidean space evolving by mean curvature, with mean curvature $H>0$ during the flow. Fix $a\geq T$ and set…

Harnack inequality conjecture. A Harnack inequality should hold whenever

Brezis–Merle sup-plus-inf conjecture. Under these assumptions, the displayed estimate should hold uniformly in $n$. This is the singular two-dimensional analogue of sup-plus-inf in…

Let $u$ and $v$ be real-valued harmonic functions in a domain $\Omega\subset\mathbb R^n$ with the same zero set $Z(u)=Z(v)=Z$, and let $f=u/v$. For a compact subset…

Let $u$ be a solution of the ultraparabolic equation considered in Theorem 1.1, and suppose that the theorem's other hypotheses hold. Boundedness-removal conjecture. Theorem 1.1 st…

Let $V$ be a real vector space of dimension $n+1$. Let $K\subset Sym^2_B(\Lambda^2V^*)$ be a closed, convex, $GL(V)$-invariant cone containing all weakly positive-definite curvatur…