Relative heat content asymptotics for sub-Riemannian manifolds
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2021
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| _version_ | 1866913572103127040 |
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| author | Agrachev, Andrei Rizzi, Luca Rossi, Tommaso |
| author_facet | Agrachev, Andrei Rizzi, Luca Rossi, Tommaso |
| contents | The relative heat content associated with a subset $Ω\subset M$ of a sub-Riemannian manifold, is defined as the total amount of heat contained in $Ω$ at time $t$, with uniform initial condition on $Ω$, allowing the heat to flow outside the domain. In this work, we obtain a fourth-order asymptotic expansion in square root of $t$ of the relative heat content associated with relatively compact non-characteristic domains. Compared to the classical heat content that we studied in [Rizzi, Rossi - J. Math. Pur. Appl., 2021], several difficulties emerge due to the absence of Dirichlet conditions at the boundary of the domain. To overcome this lack of information, we combine a rough asymptotic for the temperature function at the boundary, coupled with stochastic completeness of the heat semi-group. Our technique applies to any (possibly rank-varying) sub-Riemannian manifold that is globally doubling and satisfies a global weak Poincaré inequality, including in particular sub-Riemannian structures on compact manifolds and Carnot groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2110_03926 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Relative heat content asymptotics for sub-Riemannian manifolds Agrachev, Andrei Rizzi, Luca Rossi, Tommaso Analysis of PDEs Differential Geometry Functional Analysis 35R01, 53C17, 58J60 The relative heat content associated with a subset $Ω\subset M$ of a sub-Riemannian manifold, is defined as the total amount of heat contained in $Ω$ at time $t$, with uniform initial condition on $Ω$, allowing the heat to flow outside the domain. In this work, we obtain a fourth-order asymptotic expansion in square root of $t$ of the relative heat content associated with relatively compact non-characteristic domains. Compared to the classical heat content that we studied in [Rizzi, Rossi - J. Math. Pur. Appl., 2021], several difficulties emerge due to the absence of Dirichlet conditions at the boundary of the domain. To overcome this lack of information, we combine a rough asymptotic for the temperature function at the boundary, coupled with stochastic completeness of the heat semi-group. Our technique applies to any (possibly rank-varying) sub-Riemannian manifold that is globally doubling and satisfies a global weak Poincaré inequality, including in particular sub-Riemannian structures on compact manifolds and Carnot groups. |
| title | Relative heat content asymptotics for sub-Riemannian manifolds |
| topic | Analysis of PDEs Differential Geometry Functional Analysis 35R01, 53C17, 58J60 |
| url | https://arxiv.org/abs/2110.03926 |