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Partial Differential Equations and Functional Analysis pp 73–95Cite as

Dirichlet Forms and Degenerate Elliptic Operators

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Part of the book series: Operator Theory: Advances and Applications ((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 A ↦ L 2(A), ϕ B ∈ L 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.

This work was supported by an Australian Research Council (ARC) Discovery Grant DP 0451016.

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Auckland Private Bag, 92019, Auckland, New Zealand

    A. F. M. ter Elst

  2. Centre for Mathematics and its Applications, Mathematical Sciences Institute Australian National University, Canberra, ACT, 0200, Australia

    Derek W. Robinson

  3. Department of Mathematical Sciences, New Mexico State University, P.O. Box 30001, Las Cruces, NM, 88003-8001, USA

    Adam Sikora

  4. Department of Mathematics, Nantong University Nantong, 226007, Jiangsu Province, P.R. China

    Yueping Zhu

Authors
  1. A. F. M. ter Elst
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  2. Derek W. Robinson
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  3. Adam Sikora
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  4. Yueping Zhu
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Editor information

Editors and Affiliations

  1. Technische Universiteit Delft DIAM, P.O. Box 5031, 2600 GA, Delft, The Netherlands

    Erik Koelink , Jan van Neerven , Ben de Pagter  & Guido Sweers , ,  & 

  2. Institut für Analysis und Scientific Computing, Technische Universität Wien, Wiedner Hauptstrasse 8-10 / 101, 1040, Wien, Austria

    Annemarie Luger  & Harald Woracek  & 

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Dedicated to Philippe Clément on the occasion of his retirement

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© 2006 Birkhäuser Verlag Basel/Switzerland

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ter Elst, A.F.M., Robinson, D.W., Sikora, A., Zhu, Y. (2006). Dirichlet Forms and Degenerate Elliptic Operators. In: Koelink, E., van Neerven, J., de Pagter, B., Sweers, G., Luger, A., Woracek, H. (eds) Partial Differential Equations and Functional Analysis. Operator Theory: Advances and Applications, vol 168. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7601-5_5

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