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

Given $p\in [1,\mathrm{\infty })$, 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 $({\mathcal{F}}_{t,p}{)}_{t\in (0,1)}$ to a variant of the $p$-Dirichlet energy on ${\mathbb{R}}^N$ as $t\to 0^{+}$ both in the pointwise and in the $\mathrm{\Gamma }$-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 $\mathrm{\Gamma }$-convergence and compactness of $({\mathcal{F}}_{t,p}{)}_{t\in (0,1)}$ when the limit $p$-energy is of non-local type. Finally, we apply our results to provide asymptotic formulas in the pointwise and $\mathrm{\Gamma }$-sense for heat content-type energies both in the local and non-local settings.