7 problems

Fix an interval $T\subset\mathbb{R}$, and let $R\in\Omega\mathcal{C}(T)$ be a process. Let $p:\mathbb{R}^+\to\mathbb{R}^+$ be an increasing function with $p(0)=0$, and let $\psi$ b…

Regularity-extension conjecture. These regularity results can be extended to general Kolmogorov–Fokker–Planck operators with similar underlying geometric structures.

Consider the multiscale elliptic equation referenced as, with forcing term $f=0$, and let $(\varepsilon_i)_{1\le i\le n}$ be the oscillation scales, without imposing any scale-sepa…

Let $f:[0,T]\times\mathbb R^3\to[0,\infty)$ be a classical solution of the space-homogeneous Landau equation with the Landau-Coulomb operator $Q$, and let $f(0,v)=f_0(v)$. The Land…

Armstrong–Silvestre–Smart conjecture. The dependence of $\epsilon$ on $\Lambda$ in the interior $W^{2,\,\epsilon}$ estimate should be linear in $\Lambda^{-1}$, equivalently…

Armstrong–Silvestre–Smart's conjecture. The optimal exponent in this estimate is

Let $D\subset\mathds{R}^d$ be a bounded open set, and let $(\mu(x,\mathord{\mathrm{d}} y))_{x\in D}$ be a family of measures defining a nonlocal operator. Suppose that, for some…