标题： BBM公式的基本条件和热含量型能量的渐近性质 显示英文标题

标题： Sharp conditions for the BBM formula and asymptotics of heat content-type energies

摘要： 给定$p\in[1,\infty)$，我们提供了非负可测核$(\rho_t)_{t\in(0,1)}$的充分必要条件，以确保相关的Bourgain-Brezis-Mironescu (BBM) 能量$(\mathscr{F}_{t,p})_{t\in(0,1)}$在点态和$\Gamma$的意义下，当$t\to0^+$时收敛到$p$-Dirichlet 能量在$\mathbb R^N$上的一个变体。 我们还制定了关于$(\rho_t)_{t\in(0,1)}$的充分条件，以在$L^p(\mathbb R^N)$中产生具有有界 BBM 能量的序列的局部紧性。 此外，我们给出了关于$(\rho_t)_{t\in(0,1)}$的充分条件，以推导出当极限$p$-能量为非局部类型时，$(\mathscr{F}_{t,p})_{t\in(0,1)}$的逐点和$\Gamma$-收敛性及紧性。 最后，我们将我们的结果应用于提供在局部和非局部设置下热含量型能量的逐点和$\Gamma$-意义下的渐近公式。 显示英文摘要

摘要： Given $p\in[1,\infty)$, we provide sufficient and necessary conditions on the non-negative measurable kernels $(\rho_t)_{t\in(0,1)}$ ensuring convergence of the associated Bourgain-Brezis-Mironescu (BBM) energies $(\mathscr{F}_{t,p})_{t\in(0,1)}$ to a variant of the $p$-Dirichlet energy on $\mathbb R^N$ as $t\to0^+$ both in the pointwise and in the $\Gamma$-sense. We also devise sufficient conditions on $(\rho_t)_{t\in(0,1)}$ yielding local compactness in $L^p(\mathbb R^N)$ of sequences with bounded BBM energy. Moreover, we give sufficient conditions on $(\rho_t)_{t\in(0,1)}$ implying pointwise and $\Gamma$-convergence and compactness of $(\mathscr{F}_{t,p})_{t\in(0,1)}$ when the limit $p$-energy is of non-local type. Finally, we apply our results to provide asymptotic formulas in the pointwise and $\Gamma$-sense for heat content-type energies both in the local and non-local settings.