19 problems

Anisotropic-kernel convergence conjecture. It should be possible to generalize the aforementioned nonlocal-to-local convergence results to anisotropic kernels.

Consider sequences of parabolic nonlocal operators of the form … In the local setting, when the sequence of matrix-valued functions is independent of time, the parabolic $H$-limit…

A sequence of monotone operators in fractional divergence form is considered, with the associated fractional fluxes and gradients understood in the setting developed in the paper.…

Sharpness conjecture. The interval $p\in(2,6)$ is sharp in this mixed local-nonlocal setting; in particular, as in the known local and nonlocal limiting cases, the system should ad…

Let $2\sigma\in(0,2)$ and let $\mathcal{L}$ be a non-degenerate nonlocal operator satisfying condition (LC), let $F$ satisfy the non-degeneracy and structural assumptions, and let…

Consider the weak Harnack inequality and Hölder continuity results for the solutions of the one-obstacle, two-obstacle and N-membranes problems associated with the nonlocal operato…

Let $A:\mathbb{R}^d\to\mathbb{R}^{d\times d}$ be a bounded, measurable and strictly elliptic matrix such that … for some $a_*,a^*>0$, for every $x\in\mathbb{R}^d$ and all…

Let $\Omega\subset\mathbb R^d$ be a bounded Euclidean domain, and let $g$ and $h$ be functions supported in $\Omega$. Consider the restricted fractional Laplacian, whose nonlocal t…

Let $n$ and $s$ be the dimension and fractional-order parameters, let $\boldsymbol{x},\boldsymbol{y}$ be points in the underlying Euclidean space, and let $\lambda$ be the temperin…

Let $a$ be a kernel that decays sufficiently fast at infinity, and define … Let $\xi_\sigma$ be the rescaled support parameter, $\overline{w}_\sigma$ the rescaled profile,…

Let $a$ be a sufficiently regular kernel, and define … Let $\xi_\sigma$ be the rescaled support parameter, $\overline{w}_\sigma$ the rescaled profile, $\overline{\chi}$ the indicat…

Heat-kernel comparability conjecture. The heat kernel of $X$ is comparable to that of $Z$. The conjecture proposes robustness of heat-kernel estimates under comparable perturbation…

The method of functional multi-vectors is used to check the Jacobi identity and compatibility of Hamiltonian operators. Applicability conjecture. This method should be applicable t…

Persistence conjecture. The small effect observed for the quartic potential persists for these higher-order anharmonic non-local oscillators. The conjecture is supported by numeric…

Assume $d\geqslant 2$, that the coefficient $a(t)$ or $A(s)$ is positive and locally Lipschitz continuous, and that $V(x)$ is a radial, locally bounded confining potential. Let…

Let $L+V(x)=\psi(-\Delta)+V(x)$ be a non-local Schrödinger operator with confining potential $V$, and let $f_n$ be its $n$-th eigenfunction. A nodal part is a connected component o…

Olver's conjecture. P. Olver's method works for nonlocal operators.

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…

Mityagin's quartic-model conjecture. The constant $L$ should coincide with the lowest eigenvalue of