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21
Sobolev Spaces, Laplacian, And Heat Kernel On Alexandrov Spaces
, 1998
"... . We prove the compactness of the imbedding of the Sobolev space W 1;2 0 (\Omega\Gamma into L 2(\Omega\Gamma for any relatively compact open subset\Omega of an Alexandrov space. As a corollary, the generator induced from the Dirichlet (energy) form has discrete spectrum. The generator can be appr ..."
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Cited by 17 (6 self)
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. We prove the compactness of the imbedding of the Sobolev space W 1;2 0 (\Omega\Gamma into L 2(\Omega\Gamma for any relatively compact open subset\Omega of an Alexandrov space. As a corollary, the generator induced from the Dirichlet (energy) form has discrete spectrum. The generator can be approximated by the Laplacian induced from the DCstructure on the Alexandrov space. We also prove the existence of the locally Holder continuous Dirichlet heat kernel. 1. Introduction Consider a family M of ndimensional closed Riemannian manifolds with a uniform lower bound of sectional curvature and a uniform upper bound of diameter for a fixed n 2 N . In order to investigate various properties of manifolds in M, it is very useful to study its closure M with respect to the GromovHausdorff distance dGH , which is compact by the Gromov compactness theorem [15]. Since the closure M consists of Alexandrov spaces introduced in [2], the study of Alexandrov spaces is nowadays an important topic i...
The AllegrettoPiepenbrink Theorem for Strongly Local Dirichlet Forms
 DOCUMENTA MATH.
, 2009
"... The existence of positive weak solutions is related to spectral information on the corresponding partial differential operator. ..."
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Cited by 7 (4 self)
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The existence of positive weak solutions is related to spectral information on the corresponding partial differential operator.
A dual characterization of length spaces with application to Dirichlet metric spaces
"... We show that under minimal assumptions, the intrinsic metric induced by a strongly local Dirichlet form induces a length space. A main input is a dual characterization of length spaces in terms of the property that the 1Lipschitz functions form a sheaf. ..."
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Cited by 4 (3 self)
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We show that under minimal assumptions, the intrinsic metric induced by a strongly local Dirichlet form induces a length space. A main input is a dual characterization of length spaces in terms of the property that the 1Lipschitz functions form a sheaf.
I.: Generalized eigenfunctions and spectral theory for strongly local Dirichlet forms
"... Abstract. We present an introduction to the framework of strongly local Dirichlet forms and discuss connections between the existence of certain generalized eigenfunctions and spectral properties within this framework. The range of applications is illustrated by a list of examples. ..."
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Cited by 4 (3 self)
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Abstract. We present an introduction to the framework of strongly local Dirichlet forms and discuss connections between the existence of certain generalized eigenfunctions and spectral properties within this framework. The range of applications is illustrated by a list of examples.
Sobolev inequalities in familiar and unfamiliar settings
 In S. Sobolev Centenial Volumes, (V. Maz’ja, Ed
"... Abstract The classical Sobolev inequalities play a key role in analysis in Euclidean spaces and in the study of solutions of partial differential equations. In fact, they are extremely flexible tools and are useful in many different settings. This paper gives a glimpse of assortments of such applica ..."
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Cited by 3 (1 self)
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Abstract The classical Sobolev inequalities play a key role in analysis in Euclidean spaces and in the study of solutions of partial differential equations. In fact, they are extremely flexible tools and are useful in many different settings. This paper gives a glimpse of assortments of such applications in a variety of contexts. 1
P.: Heat kernel estimates for the ¯ ∂Neumann problem on Gmanifolds
"... Abstract. We prove heat kernel estimates for the ¯ ∂Neumann Laplacian □ acting in spaces of differential forms over noncompact manifolds with a Lie group symmetry and compact quotient. We also relate our results to those for an associated LaplaceBeltrami operator on functions. Dedicated to Barry S ..."
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Cited by 3 (2 self)
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Abstract. We prove heat kernel estimates for the ¯ ∂Neumann Laplacian □ acting in spaces of differential forms over noncompact manifolds with a Lie group symmetry and compact quotient. We also relate our results to those for an associated LaplaceBeltrami operator on functions. Dedicated to Barry Simon on his 65 th birthday Contents
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
application to Dirichlet metric spaces
, 903
"... A dual characterization of length spaces with ..."