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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Heat kernels on metric measure spaces and an application to semilinear elliptic equations


Authors: Alexander Grigor'yan, Jiaxin Hu and Ka-Sing Lau
Journal: Trans. Amer. Math. Soc. 355 (2003), 2065-2095
MSC (2000): Primary 60J35; Secondary 28A80, 35J60
Published electronically: January 10, 2003
MathSciNet review: 1953538
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Abstract: We consider a metric measure space $(M,d,\mu )$ and a heat kernel $ p_{t}(x,y)$ on $M$ satisfying certain upper and lower estimates, which depend on two parameters $\alpha $ and $\beta $. We show that under additional mild assumptions, these parameters are determined by the intrinsic properties of the space $(M,d,\mu )$. Namely, $\alpha $ is the Hausdorff dimension of this space, whereas $\beta $, called the walk dimension, is determined via the properties of the family of Besov spaces $W^{\sigma ,2}$ on $M$. Moreover, the parameters $\alpha $ and $ \beta $ are related by the inequalities $2\leq \beta \leq \alpha +1$.

We prove also the embedding theorems for the space $W^{\beta /2,2}$, and use them to obtain the existence results for weak solutions to semilinear elliptic equations on $M$ of the form

\begin{displaymath}-\mathcal{L}u+f(x,u)=g(x), \end{displaymath}

where $\mathcal{L}$ is the generator of the semigroup associated with $p_{t}$.

The framework in this paper is applicable for a large class of fractal domains, including the generalized Sierpinski carpet in ${\mathbb{R}^{n}}$.


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Additional Information

Alexander Grigor'yan
Affiliation: Department of Mathematics, Imperial College, London, SW7 2BZ, United Kingdom and The Institute of Mathematical Sciences, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong
Email: a.grigoryan@ic.ac.uk

Jiaxin Hu
Affiliation: Department of Mathematical Sciences, Tsinghua University, Beijing 100084 China and Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong
Email: jxhu@math.tsinghua.edu.cn

Ka-Sing Lau
Affiliation: Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong
Email: kslau@math.cuhk.edu.hk

DOI: http://dx.doi.org/10.1090/S0002-9947-03-03211-2
PII: S 0002-9947(03)03211-2
Received by editor(s): July 23, 2002
Published electronically: January 10, 2003
Additional Notes: The first author was partially supported by a visiting grant of the Institute of Mathematical Sciences of CUHK (the Chinese University of Hong Kong). The second author was supported by a Postdoctoral Fellowship from CUHK. The third author was partially supported by a HKRGC grant at CUHK
Article copyright: © Copyright 2003 by A. Grigor'yan, J. Hu, and K.-S. Lau