Inserted: 11 oct 2021
The relative heat content associated with a subset Ω⊂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.
Keywords: sub-Riemannian geometry, Asymptotic Analysis, Relative heat content