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

Abstract

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

38 pages. New Remark 3.2 added. Minor errors and typos corrected

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