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

TL;DR

This paper establishes precise mathematical conditions for the convergence of certain energy functionals to a Dirichlet energy in both local and non-local settings.

Contribution

The paper provides sharp conditions for BBM energy convergence and derives asymptotic formulas for heat content-type energies.

Findings

Sufficient and necessary conditions for BBM energy convergence to p-Dirichlet energy are established.

Local compactness and convergence results are derived for sequences with bounded BBM energy.

Asymptotic formulas for heat content-type energies are provided in both local and non-local settings.

Abstract

Given \documentclass[12pt]{minimal}
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\begin{document}$$p\in [1,\infty )$$\end{document}p∈[1,∞), we provide sufficient and necessary conditions on the non-negative measurable kernels \documentclass[12pt]{minimal}
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\begin{document}$$(\rho _t)_{t\in (0,1)}$$\end{document}(ρt)t∈(0,1) ensuring convergence of the associated Bourgain–Brezis–Mironescu (BBM) energies \documentclass[12pt]{minimal}
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\begin{document}$$(\mathscr {F}_{t,p})_{t\in (0,1)}$$\end{document}(Ft,p)t∈(0,1) to a variant of the p-Dirichlet energy on \documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {R}^N$$\end{document}RN as \documentclass[12pt]{minimal}
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\begin{document}$$t\rightarrow 0^+$$\end{document}t→0+ both in the pointwise and in the \documentclass[12pt]{minimal}
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\begin{document}$$\Gamma $$\end{document}Γ-sense. We also devise sufficient conditions on \documentclass[12pt]{minimal}
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\begin{document}$$(\rho _t)_{t\in (0,1)}$$\end{document}(ρt)t∈(0,1) yielding local compactness in \documentclass[12pt]{minimal}
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\begin{document}$$L^p(\mathbb {R}^N)$$\end{document}Lp(RN) of sequences with bounded BBM energy. Moreover, we give sufficient conditions on \documentclass[12pt]{minimal}
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\begin{document}$$(\rho _t)_{t\in (0,1)}$$\end{document}(ρt)t∈(0,1) implying pointwise and \documentclass[12pt]{minimal}
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\begin{document}$$\Gamma $$\end{document}Γ-convergence and equicoercivity of \documentclass[12pt]{minimal}
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\begin{document}$$({\mathscr {F}}_{t,p})_{t\in (0,1)}$$\end{document}(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 \documentclass[12pt]{minimal}
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\begin{document}$$\Gamma $$\end{document}Γ-sense for heat content-type energies both in the local and non-local settings.