Advertisement

Dirichlet Forms and Degenerate Elliptic Operators

  • A. F. M. ter Elst
  • Derek W. Robinson
  • Adam Sikora
  • Yueping Zhu
Part of the Operator Theory: Advances and Applications book series (OT, volume 168)

Abstract

It is shown that the theory of real symmetric second-order elliptic operators in divergence form on ℝd can be formulated in terms of a regular strongly local Dirichlet form irregardless of the order of degeneracy. The behavior of the corresponding evolution semigroup S t can be described in terms of a function (A, B) ↦ d(A; B) ∈ [0, ∞] over pairs of measurable subsets of ℝd. Then
$$ \left| {\left( {\varphi A,S_t \varphi B} \right)} \right| \leqslant e^{ - d(A;B)^2 (4t)^{ - 1} } ||\varphi A||2||\varphi B||2 $$
for all t > 0 and all ϕ AL 2(A), ϕ BL 2(B). Moreover S t L 2(A) ∈ L 2(A) for all t > 0 if and only if d(A;A c) = ∞ where A c denotes the complement of A.

Keywords

Heat Kernel Elliptic Operator Geodesic Distance Dirichlet Form Measurable Subset 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    L.E. Andersson, On the representation of Dirichlet forms. Ann. Inst. Fourier 25 (1975), 11–25.zbMATHMathSciNetGoogle Scholar
  2. [2]
    P. Auscher, On necessary and sufficient conditions for L p-estimates of Riesz transforms associated to elliptic operators onn and related estimates. Research report, Preprint. Univ. de Paris-Sud, 2004. Memoirs Amer. Math. Soc., to appear.Google Scholar
  3. [3]
    M. Biroli and U. Mosco, Formes de Dirichlet et estimations structurelles dans les milieux discontinus. C. R. Acad. Sci. Paris Sér. I Math. 313 (1991), 593–598.zbMATHMathSciNetGoogle Scholar
  4. [4]
    M. Biroli and U. Mosco, A Saint-Venant type principle for Dirichlet forms on discontinuous media. Ann. Mat. Pura Appl. 169 (1995).Google Scholar
  5. [5]
    A. Braides, Γ-convergence for beginners, vol. 22 of Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford, 2002.Google Scholar
  6. [6]
    N. Bouleau and F. Hirsch, Dirichlet forms and analysis on Wiener space, vol. 14 of de Gruyter Studies in Mathematics. Walter de Gruyter & Co., Berlin, 1991.zbMATHGoogle Scholar
  7. [7]
    J. Cheeger, M. Gromov, and M. Taylor, Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds. J. Differential Geom. 17 (1982), 15–53.zbMATHMathSciNetGoogle Scholar
  8. [8]
    G. Dal Maso, An introduction to Γ-convergence, vol. 8 of Progress in Nonlinear Differential Equations and their Applications. Birkhäuser Boston Inc., Boston, MA, 1993.Google Scholar
  9. [9]
    E.B. Davies, Explicit constants for Gaussian upper bounds on heat kernels. Amer. J. Math. 109 (1987), 319–333.zbMATHMathSciNetCrossRefGoogle Scholar
  10. [10]
    E.B. Davies, Heat kernel bounds, conservation of probability and the Feller property. J. Anal. Math. 58 (1992), 99–119. Festschrift on the occasion of the 70th birthday of Shmuel Agmon.zbMATHMathSciNetCrossRefGoogle Scholar
  11. [11]
    I. Ekeland and R. Temam, Convex analysis and variational problems. North-Holland Publishing Co., Amsterdam, 1976.zbMATHGoogle Scholar
  12. [12]
    A.F.M. Elst, D.W. Robinson, A. Sikora, and Y. Zhu, Second-order operators with degenerate coefficients. Research Report CASA 04-32, Eindhoven University of Technology, Eindhoven, The Netherlands, 2004.Google Scholar
  13. [13]
    A.F.M. Elst, D.W. Robinson, and Y. Zhu, Positivity and ellipticity. Proc. Amer. Math. Soc. 134 (2006), 707–714.zbMATHMathSciNetCrossRefGoogle Scholar
  14. [14]
    M. Fukushima, Y. Oshima, and M Takeda, Dirichlet forms and symmetric Markov processes, vol. 19 of de Gruyter Studies in Mathematics. Walter de Gruyter & Co., Berlin, 1994.zbMATHGoogle Scholar
  15. [15]
    M.P. Gaffney, The conservation property of the heat equation on Riemannian manifolds. Comm. Pure Appl. Math. 12 (1959), 1–11.zbMATHMathSciNetGoogle Scholar
  16. [16]
    A. Grigor’yan, Estimates of heat kernels on Riemannian manifolds. In Spectral theory and geometry (Edinburgh, 1998), vol. 273 of London Math. Soc. Lecture Note Ser., 140–225. Cambridge Univ. Press, Cambridge, 1999.Google Scholar
  17. [17]
    M. Hino and A.J. Ramírez, Small-time Gaussian behavior of symmetric diffusion semigroups. Ann. Prob. 31 (2003), 254–1295.Google Scholar
  18. [18]
    D.S. Jerison and A. Sánchez-Calle, Estimates for the heat kernel for a sum of squares of vector fields. Ind. Univ. Math. J. 35 (1986), 835–854.zbMATHCrossRefGoogle Scholar
  19. [19]
    D.S. Jerison and A. Sánchez-Calle, Subelliptic, second order differential operators. In Berenstein, C. A., ed., Complex analysis III, Lecture Notes in Mathematics 1277. Springer-Verlag, Berlin etc., 1987, 46–77.Google Scholar
  20. [20]
    J. Jost, Nonlinear Dirichlet forms. In New directions in Dirichlet forms, vol. 8 of AMS/IP Stud. Adv. Math., 1–47. Amer. Math. Soc., Providence, RI, 1998.Google Scholar
  21. [21]
    T. Kato, Perturbation theory for linear operators. Second edition, Grundlehren der mathematischen Wissenschaften 132. Springer-Verlag, Berlin etc., 1984.zbMATHGoogle Scholar
  22. [22]
    Z.M. Ma and M. Röckner, Introduction to the theory of (non symmetric) Dirichlet Forms. Universitext. Springer-Verlag, Berlin etc., 1992.zbMATHGoogle Scholar
  23. [23]
    U. Mosco, Composite media and asymptotic Dirichlet forms. J. Funct. Anal. 123 (1994), 368–421.zbMATHMathSciNetCrossRefGoogle Scholar
  24. [24]
    M. Reed and B. Simon, Methods of modern mathematical physics IV. Analysis of operators. Academic Press, New York etc., 1978.zbMATHGoogle Scholar
  25. [25]
    J.-P. Roth, Formule de représentation et troncature des formes de Dirichlet surm. In Séminaire de Théorie du Potentiel de Paris, No. 2, Lect. Notes in Math. 563, 260–274. Springer Verlag, Berlin, 1976.Google Scholar
  26. [26]
    B. Schmuland, On the local property for positivity preserving coercive forms. In Z.M. Ma and M. Röckner, eds., Dirichlet forms and stochastic processes. Walter de Gruyter & Co., Berlin, 1995, 345–354. Papers from the International Conference held in Beijing, October 25–31, 1993, and the School on Dirichlet Forms, held in Beijing, October 18–24, 1993.Google Scholar
  27. [27]
    A. Sikora, Riesz transform, Gaussian bounds and the method of wave equation. Math. Z. 247 (2004), 643–662.zbMATHMathSciNetCrossRefGoogle Scholar
  28. [28]
    B. Simon, Ergodic semigroups of positivity preserving self-adjoint operators. J. Funct. Anal. 12 (1973), 335–339.zbMATHCrossRefGoogle Scholar
  29. [29]
    B. Simon, Lower semicontinuity of positive quadratic forms. Proc. Roy. Soc. Edinburgh Sect. A 79 (1977), 267–273.zbMATHMathSciNetGoogle Scholar
  30. [30]
    B. Simon, A canonical decomposition for quadratic forms with applications to monotone convergence theorems. J. Funct. Anal. 28 (1978), 377–385.zbMATHMathSciNetCrossRefGoogle Scholar
  31. [31]
    K.-T. Sturm, Analysis on local Dirichlet spaces. II. Upper Gaussian estimates for the fundamental solutions of parabolic equations. Osaka J. Math. 32 (1995), 275–312.zbMATHMathSciNetGoogle Scholar
  32. [32]
    K.-T. Sturm, The geometric aspect of Dirichlet forms. In New directions in Dirichlet forms, vol. 8 of AMS/IP Stud. Adv. Math., 233–277. Amer. Math. Soc., Providence, RI, 1998.Google Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2006

Authors and Affiliations

  • A. F. M. ter Elst
    • 1
  • Derek W. Robinson
    • 2
  • Adam Sikora
    • 3
  • Yueping Zhu
    • 4
  1. 1.Department of MathematicsUniversity of Auckland Private BagAucklandNew Zealand
  2. 2.Centre for Mathematics and its ApplicationsMathematical Sciences Institute Australian National UniversityCanberraAustralia
  3. 3.Department of Mathematical SciencesNew Mexico State UniversityLas CrucesUSA
  4. 4.Department of MathematicsNantong University NantongJiangsu ProvinceP.R. China

Personalised recommendations