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)


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.


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.


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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

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