Definition: A family of functionals $\{F_n: X\to\bar{\mathbb R}\}$ on a metric space $X$ is said to be equi-coercive if, for every $\alpha \in \mathbb{R}$, there is a compact set $K_\alpha$ of $X$ such that for all $n$ we have $$\{F_n \leq \alpha\} \subseteq K_\alpha.$$ Question: If $\{F_n\}$ is ...

This is a soft question, I guess. $\Gamma$-convergence is a notion of convergence of functionals so that if $F_n$ $\Gamma$-converges to $F$, then cluster points of $\arg\inf F_n$ are minimizers of $F$. This is especially helpful if you want to minimize $F$ but find it easier to minimize $F_n$. Ho...

Consider the following result（$d$ denotes the dimensions and $0<t<T$） $$c\left(\sum_{j=0}^\infty\frac{\Gamma^j(1-\kappa)}{\Gamma((j+1)(1-\kappa))}t^{j(1-\kappa)-\kappa}\right)^{\frac{1}{2}}\leq c t^{-\frac{\kappa}{2}}, \text{ where } \frac{\kappa}{2}=\frac{d}{\alpha}-\frac{d}{2q\alpha}, \alpha+d\...