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

TL;DR

The article develops sharp necessary and sufficient conditions on non-negative kernels $(\rho_t)$ ensuring that BBM energies $(\mathscr{F}_{t,p})$ converge to a $p$-Dirichlet-type energy on $\mathbb{R}^N$ as $t\to0^+$, in both pointwise and $\Gamma$-sense, and proves accompanying compactness results. It introduces a non-local limit framework $(\mathscr G_p^{\mu,\nu})$ and shows that, under a maximal-rank condition, coercivity and equi-coercivity hold, with convergence to local or non-local limit energies depending on the regime; this is then specialized to special kernel families to obtain local convergence results. The paper applies the framework to heat-type energies, deriving explicit small-time asymptotics for classical and fractional heat kernels, including heat-content expansions, and extends the analysis to Hilbert spaces with alternative proofs via semigroup theory or Fourier transform. Overall, the work unifies BBM-type limits across local and non-local settings, provides sharp criteria for convergence and compactness, and yields precise heat-content-type asymptotics with broad applicability to non-local Sobolev spaces and anisotropic kernels.

Abstract

Given $p\in[1,\infty)$, we provide sufficient and necessary conditions on the non-negative measurable kernels $(ρ_t)_{t\in(0,1)}$ 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 $Γ$-sense. We also devise sufficient conditions on $(ρ_t)_{t\in(0,1)}$ yielding local compactness in $L^p(\mathbb R^N)$ of sequences with bounded BBM energy. Moreover, we give sufficient conditions on $(ρ_t)_{t\in(0,1)}$ implying pointwise and $Γ$-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 $Γ$-sense for heat content-type energies both in the local and non-local settings.

Theorems & Definitions (57)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Lemma 2.1