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Heat kernel estimates for critical fractional diffusion operators

Longjie Xie, Xicheng Zhang (2014)

Studia Mathematica

We construct the heat kernel of the 1/2-order Laplacian perturbed by a first-order gradient term in Hölder spaces and a zero-order potential term in a generalized Kato class, and obtain sharp two-sided estimates as well as a gradient estimate of the heat kernel, where the proof of the lower bound is based on a probabilistic approach.

Heat kernel estimates for the Dirichlet fractional Laplacian

Zhen-Qing Chen, Panki Kim, Renming Song (2010)

Journal of the European Mathematical Society

We consider the fractional Laplacian $- {(- Δ)}^{α / 2}$ on an open subset in $ℝ^{d}$ with zero exterior condition. We establish sharp two-sided estimates for the heat kernel of such a Dirichlet fractional Laplacian in $C^{1, 1}$ open sets. This heat kernel is also the transition density of a rotationally symmetric $α$ -stable process killed upon leaving a $C^{1, 1}$ open set. Our results are the first sharp twosided estimates for the Dirichlet heat kernel of a non-local operator on open sets.

Hölder continuity of solutions of second-order non-linear elliptic integro-differential equations

Guy Barles, Emmanuel Chasseigne, Cyril Imbert (2011)

Journal of the European Mathematical Society

This paper is concerned with the Hölder regularity of viscosity solutions of second-order, fully non-linear elliptic integro-differential equations. Our results rely on two key ingredients: first we assume that, at each point of the domain, either the equation is strictly elliptic in the classical fully non-linear sense, or (and this is the most original part of our work) the equation is strictly elliptic in a non-local non-linear sense we make precise. Next we impose some regularity and growth...

Displaying 1 – 3 of 3

Heat kernel estimates for critical fractional diffusion operators

Heat kernel estimates for the Dirichlet fractional Laplacian

Hölder continuity of solutions of second-order non-linear elliptic integro-differential equations

Currently displaying 1 – 3 of 3