Accepted Paper

Inserted: 20 feb 2025
Last Updated: 7 nov 2025

Journal: Arch. Ration. Mech. Anal.
Year: 2025

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 $(\mathscr F_{t,p})_{t\in(0,1)}$ 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.

Keywords: Gamma-convergence, Heat kernel, Compactness, Relative heat content, non-local functionals, BBM formula, Sobolev and $BV$ functions, Hilbert spaces

Download:

• Gennaioli, Stefani - Sharp conditions for the BBM formula and asymptotics of heat content-type energies