8 problems

Let the conductivity problem have inclusions of arbitrary shape in any spatial dimension, with interface parameter $gamma$ bounded away from zero. FJKL's stress-shielding conjectur…

Huang's conjecture. The matrix differential Harnack estimate should hold for this hypoelliptic operator.

Let $\Omega$ be a convex domain, and let the torsion function on $\Omega$ be the solution of the relevant Dirichlet problem. The largest inscribed circle of $\Omega$ intersects the…

Let $K$ be the best constant in the Heisenberg gradient estimates … for all $q\geqslant 1$ and $f\in C_{0}^{\infty}\left(\mathbb{G}\times\mathbb{R}^{3}\right)$, where $P_t$ is the…

Let $(\sigma_k)_{k\in\mathbb{Z}^2_0}$ be the family of vector fields on $\mathbb{T}^2$ defined, for $\gamma\geq 2$, by … Assume that the associated gradient estimate is the estimat…

Let $U^*_{\alpha}$ be the final blowup profile associated with the constructed solution, and let $c^*$ be a positive constant. The profile itself satisfies a logarithmic asymptotic…

Let $u0$, let $h0$ be general smooth initial data, and let $h=h(t,x)$ be the corresponding solution to the regularized thin film epitaxy equation … Here $abla h$ denotes the spat…

Endpoint $L^1$ gradient-bound conjecture. The same inequality should hold for $p=1$, namely, there should exist a constant $C_1>1$ such that for every smooth…