Sharp Conditions for the BBM Formula and Asymptotics of Heat Content-Type Energies

Given $p\in [1,\infty )$ , we provide sufficient and necessary conditions on the non-negative measurable kernels ${\left({\rho }_t\right)}_{t\in (0,1)}$ ensuring convergence of the associated Bourgain-Brezis-Mironescu (BBM) energies ${\left(F_{t,p}\right)}_{t\in (0,1)}$ to a variant of the p-Dirichlet energy on $R^N$ as $t\to 0^{+}$ both in the pointwise and in the $\Gamma$ -sense. We also devise sufficient conditions on ${\left({\rho }_t\right)}_{t\in (0,1)}$ yielding local compactness in $L^p\left(R^N\right)$ of sequences with bounded BBM energy. Moreover, we give sufficient conditions on ${\left({\rho }_t\right)}_{t\in (0,1)}$ implying pointwise and $\Gamma$ -convergence and equicoercivity of ${\left(F_{t,p}\right)}_{t\in (0,1)}$ 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.