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18
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 ..."
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Cited by 47 (24 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.
Stability results for Harnack inequalities
, 2004
"... We develop new techniques for proving uniform elliptic and parabolic Harnack inequalities on weighted Riemannian manifolds. In particular, we prove the stability of the Harnack inequalities under certain nonuniform changes of the weight. We also prove necessary and sufficient conditions for the Har ..."
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Cited by 25 (2 self)
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We develop new techniques for proving uniform elliptic and parabolic Harnack inequalities on weighted Riemannian manifolds. In particular, we prove the stability of the Harnack inequalities under certain nonuniform changes of the weight. We also prove necessary and sufficient conditions for the Harnack inequalities to hold on complete noncompact manifolds having nonnegative Ricci curvature outside a compact set and a finite first Betti number or just having asymptotically nonnegative sectional curvature.
Estimates on the Dirichlet heat kernel of domains above the graphs of bounded C 1,1 functions
"... Abstract. Suppose that D is the domain in Rd, d 3, above the graph of a bounded C1;1 function : Rd1! R and that pD(t; x; y) is the Dirichlet heat kernel in D. In this paper we show that there exist positive constants C1, C2, C3 and C4 such that for all t> 0 and x; y 2 D, C1( (x)(y) ..."
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Cited by 17 (11 self)
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Abstract. Suppose that D is the domain in Rd, d 3, above the graph of a bounded C1;1 function : Rd1! R and that pD(t; x; y) is the Dirichlet heat kernel in D. In this paper we show that there exist positive constants C1, C2, C3 and C4 such that for all t> 0 and x; y 2 D, C1( (x)(y)
Critical points of Green’s functions on complete manifolds
 J. Differential Geom
"... We prove that the number of critical points of a Li–Tam Green’s function on a complete open Riemannian surface of finite type admits a topological upper bound, given by the first Betti number of the surface. In higher dimensions, we show that there are no topological upper bounds on the number of c ..."
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Cited by 8 (7 self)
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We prove that the number of critical points of a Li–Tam Green’s function on a complete open Riemannian surface of finite type admits a topological upper bound, given by the first Betti number of the surface. In higher dimensions, we show that there are no topological upper bounds on the number of critical points by constructing, for each nonnegative integer N, a Riemannian manifold diffeomorphic to Rn (n> 3) whose minimal Green’s function has at least N nondegenerate critical points. Variations on the method of proof of the latter result yield contractible nmanifolds whose minimal Green’s functions have level sets diffeomorphic to any fixed codimension 1 compact submanifold of Rn. 1.
Geometrical and topological aspects of Electrostatics on Riemannian manifolds
 J. Geom. Phys
"... We study some geometrical and topological properties of the electric fields created by point charges on Riemannian manifolds from the viewpoint of the theory of dynamical systems. We provide a thorough description of the structure of the basin boundary and its connection with the topology of the ma ..."
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Cited by 7 (4 self)
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We study some geometrical and topological properties of the electric fields created by point charges on Riemannian manifolds from the viewpoint of the theory of dynamical systems. We provide a thorough description of the structure of the basin boundary and its connection with the topology of the manifold, and characterize the spaces in which the orbits of the electric field are geodesics. We also consider symmetries of electric fields on manifolds, particularly on spaces of constant curvature.
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 ..."
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Cited by 4 (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
Movement of hot spots on the exterior domain of a ball
"... We consider the movement of the maximum points of the solutions of the CauchyNeumann problem and the CauchyDirichlet of the heat equation, ∂tu = ∆u in Ω × (0,∞), ∂νu = 0 on ∂Ω × (0,∞), ..."
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Cited by 3 (1 self)
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We consider the movement of the maximum points of the solutions of the CauchyNeumann problem and the CauchyDirichlet of the heat equation, ∂tu = ∆u in Ω × (0,∞), ∂νu = 0 on ∂Ω × (0,∞),
The heat kernel and its estimates
, 2008
"... 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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Cited by 3 (0 self)
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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.
THE FUJITA EXPONENT FOR SEMILINEAR HEAT EQUATIONS WITH QUADRATICALLY DECAYING POTENTIAL OR IN AN EXTERIOR DOMAIN ROSS
"... Abstract. Consider the equation ..."
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