Sharp conditions for the BBM formula and asymptotics of heat content-type energies
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| Format: | Preprint |
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2025
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| _version_ | 1866908634747764736 |
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| author | Gennaioli, Luca Stefani, Giorgio |
| author_facet | Gennaioli, Luca Stefani, Giorgio |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_14655 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sharp conditions for the BBM formula and asymptotics of heat content-type energies Gennaioli, Luca Stefani, Giorgio Analysis of PDEs Functional Analysis Primary 46E35. Secondary 26A33, 26D10, 35K08 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. |
| title | Sharp conditions for the BBM formula and asymptotics of heat content-type energies |
| topic | Analysis of PDEs Functional Analysis Primary 46E35. Secondary 26A33, 26D10, 35K08 |
| url | https://arxiv.org/abs/2502.14655 |