Neumann and Dirichlet Heat Kernels in Inner Uniform Domains
By L. Saloff‐Coste, Pavel Gyrya. Astérisque, 2011
154 citations recorded by the source corpus.
— This monograph focuses on the heat equation with either the Neumann or the Dirichlet boundary condition in unbounded domains in Euclidean space, Rie-mannian manifolds, and in the more general context of certain regular local Dirichlet spaces. In works by A. Grygor'yan, L. Saloff-Coste and K-T. Sturm, the equivalence between – The parabolic Harnack inequality – The two-sided Gaussian heat kernel estimate – The Poincaré inequality and the volume doubling property is established in a very general context. We use this result to provide precise two-sided heat kernel estimates in a large class of domains described in terms of their inner intrinsic metric and called inner (or intrinsically) uniform domains. Perhaps surprisingly , we treat both the Neumann boundary condition and the Dirichlet boundary condition using essentially the same approach albeit with the additional help of a Doob's h-transform in the case of Dirichlet boundary condition. The main results are new even when applied to Euclidean domains with smooth boundary where they capture the global effect of the condition of inner uniformity as, for instance, in the case of domains that are complements of a convex set in Euclidean space.
Subfields: Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/f7ea2885d782c3f08ab77d53bba1559c1bb296f5
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Mathematical Physics
- M. Ryznar
- T. Grzywny
- Krzysztof Bogdan
- Xavier Ros-Oton
- Tomasz Jakubowski
- Paweł Sztonyk
- Renming Song
- Zhen-Qing Chen
Citations in the corpus
- Dirichlet heat kernel for unimodal Lévy processes (2014, background): “6. In general we strive to control comparability constants because they may be important in scaling arguments and applications to more general Markov processes. In passing we also refer the reader to [25] for heat kernel estimates of unbounded domains for second-order elliptic differential operators.”