8 problems

Let $P$ be a critical operator in a Riemannian manifold $\mathcal{M}$, and let $\varphi$ denote its ground state. Let $\mathcal{H}_P(\mathcal{M}\times\mathbb{R})$ be the cone of po…

Let $\Omega\subset\mathbb{R}^N$, with $1\leq N\leq 3$, be a bounded domain with smooth boundary, and let $u_0\in H^2\cap H_0^1(\Omega)$. Let $\lambda^{*}$ be the critical parameter…

Let $\Omega\subset\mathbb{R}^N$, with $1\leq N\leq 3$, be a bounded domain with smooth boundary, and let $u_0\in H^2\cap H_0^1(\Omega)$. Consider equation (1.1), its solution…

Kinetic Krylov–Safonov conjecture. There exist constants $\alpha>0$ and $C$, depending only on the dimension and $\lambda,\Lambda$, such that

The paper considers Assumption … can be satisfied for smooth domains. The preceding discussion establishes satisfiability for rectangular domains; the conjecture proposes the corre…

High-order accuracy conjecture. The error in the computed solution has second- and fourth-order accuracy in the maximum norm for the single- and composite-domain parabolic problems…

Consider the semilinear parabolic problem denoted by equation … z=frac{x}{sqrt{(T-t)|log(T-t)|}}. … bar{fbeta}(z)=left(p-1+b(beta)|z|^2right)^{-frac{1}{p-1}}, … has a solution…

Let $u: \mathbb{R} \times \mathbb{R}_+ \to \mathbb{R}$ solve the parabolic equation with initial data $u_0$ satisfying … where $(b_n)_{n\in\mathbb{N}}$ is strictly increasing,…