Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 ${\mathbb{R}^{n}}$.

References [Enhancements On Off] (What's this?)

  • 1. A. Ambrosetti and G. Prodi, A primer of nonlinear analysis, Cambridge University Press, 1993. MR 94f:58016
  • 2. N. Aronszajn and K. T. Smith, Theory of Bessel potentials. part 1, Ann. Inst. Fourier, Grenoble 11 (1961), 385-475. MR 26:1485
  • 3. M. T. Barlow, Diffusions on fractals, Lectures on Probability Theory and Statistics, vol. 1690, Lecture Notes Math., Springer-Verlag, 1998, pp. 1-121. MR 2000a:60148
  • 4. -, Which values of the volume growth and escape time exponents are possible for graphs?, preprint (2001).
  • 5. M. T. Barlow and R. F. Bass, Brownian motion and harmonic analysis on Sierpinski carpets, Canadian J. Math. 51 (1999), 673-744. MR 2000i:60083
  • 6. M. Biroli and U. Mosco, Sobolev and isoperimetric inequalities for Dirichlet forms on homogeneous spaces, Rend. Mat. Acc. Lincei VI (1) (1995), 37-44. MR 96i:46034a
  • 7. E. A. Carlen, S. Kusuoka, and D. W. Stroock, Upper bounds for symmetric Markov transition functions, Ann. Inst. Henri Poincaré-Probab. Statist. 23 (1987), 245-287. MR 88i:35066
  • 8. E. B. Davies, One-parameter semigroups, Academic Press, 1980. MR 82i:47060
  • 9. -, Heat kernels and spectral theory, Cambridge University Press, 1990. MR 92a:35035
  • 10. K. J. Falconer, Fractal geometry--mathematical foundation and applications, John Wiley, 1992. MR 92j:28008
  • 11. P. J. Fitzsimmons, B. M. Hambly, and T. Kumagai, Transition density estimates for Brownian motion on affine nested fractals, Commun. Math. Phys. 165 (1994), 595-620. MR 95j:60122
  • 12. M. Fukushima, Y. Oshima Y., and M. Takeda, Dirichlet forms and symmetric Markov processes, Studies in Mathematics, 19, De Gruyter, 1994. MR 96f:60126
  • 13. A. Grigor'yan, Estimates of heat kernels on Riemannian manifolds, London Math. Soc. Lecture Note Ser., vol. 273, 1999, pp. 140-225. MR 2001b:58040
  • 14. A. Grigor'yan and A. Telcs, Sub-Gaussian estimates of heat kernels on infinite graphs, Duke Math. J. 109 (2001), 451-510.
  • 15. P. Haj\lasz and P. Koskela, Sobolev met Poincaré, Mem. Amer. Math. Soc. 145 (2000), no. 688. MR 2000j:46063
  • 16. E. Hille and R. S. Phillips, Functional analysis and semi-groups, Amer. Math. Soc., 1957. MR 19:664d
  • 17. J. Hu and K. S. Lau, Riesz potentials and Laplacians on fractals, preprint (2001).
  • 18. A. Jonsson, Brownian motion on fractals and function spaces, Math. Zeit. 222 (1996), 495-504. MR 97e:60137
  • 19. I. Kuzin and S. Pohozaev, Entire solutions of semilinear elliptic equations, Birkhäuser, 1997. MR 99d:35050
  • 20. P. Li and S. T. Yau, On the parabolic kernel of the Schrödinger operator, Acta Math. 156 (1986), 153-201. MR 87f:58156
  • 21. R. A. Macías and C. Segovia, Lipschitz functions on spaces of homogeneous type, Adv. Math. 33 (1979), 257-270. MR 81c:32017a
  • 22. K. Pietruska-Pa\luba, Some function spaces related to the Brownian motion on simple nested fractals, Stochastics and Stochastics Reports 67 (1999), 267-285. MR 2000h:60078
  • 23. -, On function spaces related to fractional diffusions on $d$-sets, Stochastics and Stochastics Reports 70 (2000), 153-164. MR 2001i:60123
  • 24. P. R. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, Expository Lectures from CBMS Regional Conference held at University of Miami, Amer. Math. Soc., Providence, RI, 1986. MR 87j:58024
  • 25. A. Stós, Symmetric $\alpha$-stable processes on $d$-sets, Bull. Pol. Acad. Sci. Math. 48 (2000), 237-245. MR 2002f:60152
  • 26. A. Telcs, Random walks on graphs, electric networks and fractals, J. Probability Theory and Related Fields 82 (1989), 435-449. MR 90h:60065
  • 27. M. M. Vainberg, Variational methods for the study of nonlinear operators, San Francisco: Holden-Day, Inc., 1964. MR 31:638
  • 28. N. Th. Varopoulos, Hardy-Littlewood theory for semigroups, J. Funct. Anal. 63 (1985), 240-260. MR 87a:31011
  • 29. K. Yosida, Functional analysis, Springer-Verlag, 1980. MR 82i:46002

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 60J35, 28A80, 35J60

Retrieve articles in all journals with MSC (2000): 60J35, 28A80, 35J60

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

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

Ka-Sing Lau
Affiliation: Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong

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

American Mathematical Society