Heat Kernel Estimates for Dirichlet Fractional Laplacian
نویسندگان
چکیده
In this paper, we consider the fractional Laplacian −(−∆)α/2 on an open subset in R with zero exterior condition. We establish sharp two-sided estimates for the heat kernel of such Dirichlet fractional Laplacian in C open sets. This heat kernel is also the transition density of a rotationally symmetric α-stable process killed upon leaving a C open set. Our results are the first sharp two-sided estimates for the Dirichlet heat kernel of a non-local operator on open sets. AMS 2000 Mathematics Subject Classification: Primary 60J35, 47G20, 60J75; Secondary 47D07
منابع مشابه
Heat kernel estimates for the Dirichlet fractional Laplacian
Abstract. We consider the fractional Laplacian −(−1)α/2 on an open subset in Rd with zero exterior condition. We establish sharp two-sided estimates for the heat kernel of such a Dirichlet fractional Laplacian inC1,1 open sets. This heat kernel is also the transition density of a rotationally symmetric α-stable process killed upon leaving a C1,1 open set. Our results are the first sharp twoside...
متن کاملHeat kernel estimates for the fractional Laplacian
Explicit sharp estimates for the Green function of the Laplacian in C domains were completed in 1986 by Zhao [42]. Sharp estimates of the Green function of Lipschitz domains were given in 2000 by Bogdan [6]. Explicit qualitatively sharp estimates for the classical heat kernel in C domains were established in 2002 by Zhang [41]. Qualitatively sharp heat kernel estimates in Lipschitz domains were...
متن کاملDirichlet Heat Kernel Estimates for Fractional Laplacian with Gradient Perturbation
By Zhen-Qing Chen∗,‡, Panki Kim†,§ and Renming Song¶ University of Washington‡, Seoul National University§ and University of Illinois¶ Suppose that d ≥ 2 and α ∈ (1, 2). Let D be a bounded C open set in R and b an R-valued function on R whose components are in a certain Kato class of the rotationally symmetric α-stable process. In this paper, we derive sharp two-sided heat kernel estimates for ...
متن کاملUltracontractivity and the Heat Kernel for Schrijdinger Operators and Dirichlet Laplacians
connections between integral kernels of positivity preserving semigroups and suitable Lp contractivity properties are established. Then these questions are studied for the semigroups generated by -A + V and H,, the Dirichlet Laplacian for an open, connected region Q. As an application under a suitable hypothesis, Sobolev estimates are proved valid up to 352, of the form /n(x)1 ,< coo(x) lJHk,nl...
متن کاملRemarks on the fractional Laplacian with Dirichlet boundary conditions and applications
We prove nonlinear lower bounds and commutator estimates for the Dirichlet fractional Laplacian in bounded domains. The applications include bounds for linear drift-diffusion equations with nonlocal dissipation and global existence of weak solutions of critical surface quasi-geostrophic equations.
متن کامل