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Relative heat content
asymptotics for sub-Riemannian manifolds
Andrei Agrachev, Luca Rizzi and Tommaso Rossi |
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Vol. 17 (2024), No. 9, 2997–3037
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Abstract |
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The relative heat content associated with a subset of a sub-Riemannian manifold is defined as the total amount of heat contained in at time , with uniform initial condition on , allowing the heat to flow outside the domain. We obtain a fourth-order asymptotic expansion in the square root of of the relative heat content associated with relatively compact noncharacteristic domains. Compared to the classical heat content that was studied by Rizzi and Rossi (J. Math. Pures Appl. 148 (2021), 267–307), 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 asymptotics for the temperature function at the boundary, coupled with stochastic completeness of the heat semigroup. 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. |
Keywords
relative heat content, sub-Riemannian geometry, asymptotic
expansion
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Mathematical Subject Classification
Primary: 35R01, 53C17, 58J60
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Milestones
Received: 10 January 2022
Revised: 17 March 2023
Accepted: 31 August 2023
Published: 1 November 2024
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Authors |
| Andrei Agrachev | |
| Scuola Internazionale Superiore di
Studi Avanzati Trieste Italy |
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| Luca Rizzi | |
| Scuola Internazionale Superiore di
Studi Avanzati Trieste Italy |
Université Grenoble Alpes CNRS, Institut Fourier Grenoble France |
| Tommaso Rossi | |
| Sorbonne Université Laboratoire Jacques-Louis Lions Paris France |
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| © 2024 The Author(s), under exclusive license to MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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