Are there Harnack type inequalities and Schauder type estimates for the spectral fractional Laplacian. References are welcome.
$\begingroup$
$\endgroup$
1
-
$\begingroup$ (Spectral fractional Laplacian means the operator obtained by raising to a power the eigenspectrum in the spectral decomposition of the Laplacian) $\endgroup$– reunsCommented Sep 12, 2018 at 19:54
Add a comment
|
3 Answers
$\begingroup$
$\endgroup$
This seems like a reliable entry point to the literature: What Is the Fractional Laplacian?
The purpose of this work is two-fold: (i) to give a comprehensive report of the commonly used definitions of the fractional Laplacian and examine their differences in bounded domains, and (ii) to quantitatively assess the available numerical methods developed for each definition. Of significance is the inclusion of recent work on implementing methods for nonzero boundary value problems. This work may be of use to practitioners looking to gain insight into which fractional Laplacian definition and associated numerical methods may be appropriate for their application. We present new solvers and discuss our implementation of existing numerical methods for discretizing the fractional Laplacians.
answered May 4, 2018 at 6:06
$\begingroup$
$\endgroup$
2
A quick Google Scholar search returns "An extension problem related to the fractional Laplacian" by L. Caffarelli and L. Silvestre, an online preprint of which can be found on Arxiv,
which itself cites "Properties of the solutions of the linearized Monge-Ampère equation by L. Caffarelli and C. Gutiérrez".
However it's not clear to me that the latter actually proves an Harnack inequality for fractional operators.
-
$\begingroup$ It does prove Harnack inequality for the fractional Laplace operator, but for a different one: full-space with zero (Dirichlet) boundary condition. $\endgroup$ Commented Sep 12, 2018 at 19:07
-
$\begingroup$ That said, I realise the methods of Caffarelli and Silvestre paper extend naturally to the extension problem for the spectral fractional Laplacian. I checked the literature, and the result is given in Harnack's inequality for fractional nonlocal equations by Pablo Raúl Stinga and Chao Zhang. $\endgroup$ Commented Sep 12, 2018 at 19:45
$\begingroup$
$\endgroup$
On the probability side, this operator was extensively studied by Zhen-Qing Chen, Panki Kim, Renming Song and Zoran Vondraček. I always thought that Harnack inequality was proved in already in their first paper on that subject, Potential theory of subordinate killed Brownian motion in a domain (by R. Song and Z. Vondraček). Only now I realised it is not there.
It appears Harnack inequality was first proved by P. Kim and Ante Mimica in Harnack inequalities for subordinate Brownian motions. A general version, and much more, can be found in more recent Potential theory of subordinate killed Brownian motion.
All papers mentioned above can be found on arXiv, as well as on websites of the authors.
Regarding Schauder estimates, I do not think this is available in the probability literature. Quite likely the result can be found on the PDE side, which, unfortunately, I do not know well.
Edited: On the PDE side, Harnack inequality is proved in Harnack's inequality for fractional nonlocal equations by Pablo Raúl Stinga and Chao Zhang. Quite likely follow-ups to that paper contain regularity results you are interested in.
answered Sep 12, 2018 at 19:30