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Global heat kernel estimates for fractional Laplacians in unbounded open sets
 Probab. Theory Relat. Fields, DOI 10.1007/s0044000902560 (online first
"... In this paper, we derive global sharp heat kernel estimates for symmetric αstable processes (or equivalently, for the fractional Laplacian with zero exterior condition) in two classes of unbounded C 1,1 open sets in R d: halfspacelike open sets and exterior open sets. These open sets can be disco ..."
Abstract

Cited by 19 (13 self)
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In this paper, we derive global sharp heat kernel estimates for symmetric αstable processes (or equivalently, for the fractional Laplacian with zero exterior condition) in two classes of unbounded C 1,1 open sets in R d: halfspacelike open sets and exterior open sets. These open sets can be disconnected. We focus in particular on explicit estimates for pD(t, x, y) for all t> 0 and x, y ∈ D. Our approach is based on the idea that for x and y in D far from the boundary and t sufficiently large, we can compare pD(t, x, y) to the heat kernel in a well understood open set: either a halfspace or R d; while for the general case we can reduce them to the above case by pushing x and y inside away from the boundary. As a consequence, sharp Green functions estimates are obtained for the Dirichlet fractional Laplacian in these two types of open sets. Global sharp heat kernel estimates and Green function estimates are also obtained for censored stable processes (or equivalently, for regional fractional Laplacian) in exterior open sets.
GEOMETRIC ANALYSIS
, 2005
"... This was a talk I gave in the occasion of the seventieth anniversary of the Chinese Mathematical Society. I dedicated it in memory of my teacher S. S. Chern who passed away half a year ago. During my graduate study, I was rather free in picking research topics. I [538] worked on fundamental groups o ..."
Abstract

Cited by 3 (0 self)
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This was a talk I gave in the occasion of the seventieth anniversary of the Chinese Mathematical Society. I dedicated it in memory of my teacher S. S. Chern who passed away half a year ago. During my graduate study, I was rather free in picking research topics. I [538] worked on fundamental groups of manifolds with nonpositive curvature. But in the second year of my study, I started to look into differential equations on manifolds. While Chern did not express much opinions on this part of my research, he started to appreciate it a few years later. In fact, after Chern gave a course on Calabi’s works on affine geometry in 1972 in Berkeley, Cheng told me these inspiring lectures. By 1973, Cheng and I started to work on some problems mentioned in his lectures. We did not realize that great geometers Pogorelov, Calabi and Nirenberg were also working on them. We were excited that we solved some of the conjectures of Calabi on improper affine spheres. But soon we found out that Pogorelov [398] published it right before us by different arguments. Nevertheless our ideas are useful to handle other problems in
The heat kernel and its estimates
"... After a short survey of some of the reasons that make the heat kernel an important object of study, we review a number of basic heat kernel estimates. We then describe recent results concerning (a) the heat kernel on certain manifolds with ends, and (b) the heat kernel with Neumann or Dirichlet boun ..."
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After a short survey of some of the reasons that make the heat kernel an important object of study, we review a number of basic heat kernel estimates. We then describe recent results concerning (a) the heat kernel on certain manifolds with ends, and (b) the heat kernel with Neumann or Dirichlet boundary condition in Euclidean domains. This text is a revised version of the four lectures given by the author at the First MSJSI in Kyoto during the summer of 2008. The structure of the lectures has been mostly preserved although some material has been added, deleted, or shifted around. The goal is to present an
(1.1) THE FUJITA EXPONENT FOR SEMILINEAR HEAT EQUATIONS WITH QUADRATICALLY DECAYING POTENTIAL
, 805
"... ut = ∆u − V u + au p in R n × (0, T); u(x,0) = φ(x) ≩ 0, in R n, where p> 1, n ≥ 2, T ∈ (0, ∞], V (x) ∼ ω x  2 as x  → ∞, for some ω = 0, and a(x) is on the order x  m as x  → ∞, for some m ∈ (−∞, ∞). A solution to the above equation is called global if T = ∞. Under some additional tech ..."
Abstract
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ut = ∆u − V u + au p in R n × (0, T); u(x,0) = φ(x) ≩ 0, in R n, where p> 1, n ≥ 2, T ∈ (0, ∞], V (x) ∼ ω x  2 as x  → ∞, for some ω = 0, and a(x) is on the order x  m as x  → ∞, for some m ∈ (−∞, ∞). A solution to the above equation is called global if T = ∞. Under some additional technical conditions, we calculate a critical exponent p ∗ such that global solutions exist for p> p ∗ , while for 1 < p ≤ p ∗ , all solutions blow up in finite time. 1.