6 problems

Let $(M,g)$ be a complete Riemannian manifold, let $\Delta$ be its negative definite Laplace–Beltrami operator, and let $L^2$ positivity preservation for $-\Delta+1$ mean that … fo…

BMS conjecture. Assume that $(M,g)$ is geodesically complete. Then $(M,g)$ is $L^{2}$-positivity preserving.

Let $(M,g)$ be a complete Riemannian manifold. It is $L^2$-positivity preserving if, for every $u\in L^2(M)$, the implication … holds. BMS conjecture. If $(M,g)$ is geodesically co…

Negativity conjecture. We conjecture that

Let $(M,g)$ be a geodesically complete Riemannian manifold. It is called $L^{2}$-positivity preserving if every $u\in L^{2}(M)$ satisfying … distributionally on $M$ also satisfies…

Let $\bar{\mathbf U}_{ij}^{n+1}$ be the updated cell-average state, and let $\bar{\mathbf U}_{i,j}^n$, $\bar{\mathbf U}_{i\pm1,j}^n$, and $\bar{\mathbf U}_{i,j\pm1}^n$ denote the c…