Heat kernel estimates for the fractional Laplacian with Dirichlet conditions

By Krzysztof Bogdan, T. Grzywny, M. Ryznar. 2009

148 citations recorded by the source corpus.

We give sharp estimates for the heat kernel of the fractional Laplacian with Dirichlet condition for a general class of domains including Lipschitz domains. AMS 2000 subject classifications: Primary 60J35, 60J50; secondary 60J75, 31B25. Keywords and phrases: fractional Laplacian, Dirichlet problem, heat kernel estimate, Lipschitz domain, boundary Harnack principle.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/d81ef7a394c9335ae01868348a7904b51e28668b

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In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

The Pohozaev Identity for the Fractional Laplacian (2012, unknown): “For example, when = B1(0), the solution of { (− )su = 1 in B1(0) u = 0 in Rn\B1(0) can be computed explicitly [3,13]: u(x) = 2 −2s (n/2) ( n+2s 2 ) (1 + s) ( 1 − |x |2 )s .”
Dirichlet heat kernel for unimodal Lévy processes (2014, background): “The case of the subordinate Brownian motions with global scalings is resolved in [33] for the halfspace, and [6, 7] handle the fractional Laplacian in cones.”
Estimates of the Green Function for the Fractional Laplacian Perturbed by Gradient (2010, methodology): “In passing we also note that an approximate factorization of pD(t, x, y) for Lipschitz domains is given in [10].”
Dirichlet heat kernel estimates for Δα/2 + Δβ/2 (2010, background): “Also some ideas in [5], [10] to obtain the lower bound are adapted in this paper.”
Boundary Harnack inequality for Markov processes with jumps (2012, unknown): “The estimates depend on BHI ([24]), and reflect the fundamental decay rate in BHI ([31, 46]).”
A Priori Estimates for Fractional Nonlinear Degenerate Diffusion Equations on Bounded Domains (2013, background): “[5, 28] but γ ∈ [0, 1] may vary from the case of regular domains, the constants of the bounds depend strongly on the domain, and the distance to the boundary has to be interpreted in a suitable way.”
The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary (2012, background): “, [3, 10]; an explicit expression of the Poisson kernel for a ball [18]; and the explicit solution to problem (1.”