Closing the gap: Maz'ya-Shaposhnikova and asymptotics of fractional perimeters
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866909014712909824 |
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| author | Davoli, Elisa Fanizza, Alberto Picerni, Marco |
| author_facet | Davoli, Elisa Fanizza, Alberto Picerni, Marco |
| contents | We prove a generalization of the Maz'ya-Shaposhnikova formula in the case $p=2$ for functions that may not belong to ${L^2}(\mathbb{R}^d)$ and, thus, might not vanish at infinity. By introducing a notion of mass at infinity, we explicitly characterize the limit as $s\to0^+$ of Gagliardo seminorms localized on a bounded Lipschitz domain $Ω$. By `localized', we mean here that we account only for interactions involving at least one point in $Ω$. The identified limiting functional provides a unifying framework to link the classical Maz'ya-Shaposhnikova formula and the asymptotics of nonlocal perimeters. On the one hand, it reduces to the classical $L^2$ norm for functions that are globally integrable on $\mathbb{R}^d$. On the other hand, it recovers the pointwise limit of $s$-fractional perimeters when evaluated on characteristic functions of sets. We further show that the same functional encodes the asymptotic behavior of Gagliardo seminorms in the sense of Gamma-convergence with respect to the weak-$L^2$ topology. Finally, we provide an extension to the setting of metric measure spaces. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_03955 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Closing the gap: Maz'ya-Shaposhnikova and asymptotics of fractional perimeters Davoli, Elisa Fanizza, Alberto Picerni, Marco Analysis of PDEs Metric Geometry 46E35, 26A33, 46B20 We prove a generalization of the Maz'ya-Shaposhnikova formula in the case $p=2$ for functions that may not belong to ${L^2}(\mathbb{R}^d)$ and, thus, might not vanish at infinity. By introducing a notion of mass at infinity, we explicitly characterize the limit as $s\to0^+$ of Gagliardo seminorms localized on a bounded Lipschitz domain $Ω$. By `localized', we mean here that we account only for interactions involving at least one point in $Ω$. The identified limiting functional provides a unifying framework to link the classical Maz'ya-Shaposhnikova formula and the asymptotics of nonlocal perimeters. On the one hand, it reduces to the classical $L^2$ norm for functions that are globally integrable on $\mathbb{R}^d$. On the other hand, it recovers the pointwise limit of $s$-fractional perimeters when evaluated on characteristic functions of sets. We further show that the same functional encodes the asymptotic behavior of Gagliardo seminorms in the sense of Gamma-convergence with respect to the weak-$L^2$ topology. Finally, we provide an extension to the setting of metric measure spaces. |
| title | Closing the gap: Maz'ya-Shaposhnikova and asymptotics of fractional perimeters |
| topic | Analysis of PDEs Metric Geometry 46E35, 26A33, 46B20 |
| url | https://arxiv.org/abs/2605.03955 |