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

Given p∈[1,∞), we provide sufficient and necessary conditions on the non-negative measurable kernels (ρt)t∈(0,1) ensuring convergence of the associated Bourgain–Brezis–Mironescu (BBM) energies (Ft,p)t∈(0,1) to a variant of the p-Dirichlet energy on RN as t→0+ both in the pointwise and in the Γ-sense. We also devise sufficient conditions on (ρt)t∈(0,1) yielding local compactness in Lp(RN) of sequences with bounded BBM energy. Moreover, we give sufficient conditions on (ρt)t∈(0,1) implying pointwise and Γ-convergence and equicoercivity of (Ft,p)t∈(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 Γ-sense for heat content-type energies both in the local and non-local settings.