4 problems

Let $u$ and $v$ satisfy the hypotheses of the preceding strong Laplace Hopf-lemma conjecture, including $u_t(0,0)=0$. Finite-order vanishing conjecture. If, in addition, $u(t,0)$ a…

Strong Laplace Hopf-lemma conjecture. If, in addition, $u_t(0,0)=0$, then

Hopf-type curvature comparison problem. Is it true that either $u\equiv v$ near $(0,0)$ or $v\equiv0$ near $(0,0)$? The source describes this as a plausible variation of the Hopf L…

Let $u\ge v$ be $C^\infty$ functions of $(t,y)$ on … where $y\in\mathbb R^n$, smooth on the closure of $\Omega$, satisfying $u>0$, $v>0$, and $u_t>0$ in $\Omega$, with $u(0,y)=0$ f…