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

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- by Alexander Grigor’yan, Jiaxin Hu and Ka-Sing Lau PDF
- Trans. Amer. Math. Soc.
**355**(2003), 2065-2095

## 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{equation*} -\mathcal {L}u+f(x,u)=g(x), \end{equation*} 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 Sierpiński 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
- MR Author ID: 203816
- 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
- MR Author ID: 190087
- Email: kslau@math.cuhk.edu.hk
- 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
- © Copyright 2003 by A. Grigor’yan, J. Hu, and K.-S. Lau
- Journal: Trans. Amer. Math. Soc.
**355**(2003), 2065-2095 - MSC (2000): Primary 60J35; Secondary 28A80, 35J60
- DOI: https://doi.org/10.1090/S0002-9947-03-03211-2
- MathSciNet review: 1953538