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FLOWS AND INVARIANCE FOR DEGENERATE ELLIPTIC OPERATORS

Part of: Elliptic equations and systems

Published online by Cambridge University Press:  01 August 2011

A. F. M. TER ELST ,
DEREK W. ROBINSON  and
ADAM SIKORA
A. F. M. TER ELST*
Affiliation:
Department of Mathematics, University of Auckland, Private Bag 92019, Auckland 1142, New Zealand (email: [email protected])
DEREK W. ROBINSON
Affiliation:
Mathematical Sciences Institute, Australian National University, Canberra, ACT 0200, Australia (email: [email protected])
ADAM SIKORA
Affiliation:
Department of Mathematics, Macquarie University, Sydney, NSW 2109, Australia (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
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Abstract

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Let S be a sub-Markovian semigroup on L2(ℝd) generated by a self-adjoint, second-order, divergence-form, elliptic operator H with W1,∞(ℝd) coefficients ckl, and let Ω be an open subset of ℝd. We prove that if either C∞c(ℝd) is a core of the semigroup generator of the consistent semigroup on Lp(ℝd) for some p∈[1,∞]  or Ω has a locally Lipschitz boundary, then S leaves L2 (Ω) invariant if and only if it is invariant under the flows generated by the vector fields ∑ dl=1ckl∂l for all k. Further, for all p∈[1,2] we derive sufficient conditions on the coefficients for the core property to be satisfied. Then by combination of these results we obtain various examples of invariance in terms of boundary degeneracy both for Lipschitz domains and domains with fractal boundaries.

Keywords

degenerate elliptic operatorflowinvariance

MSC classification

Secondary: 35J70: Degenerate elliptic equations
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 90 , Issue 3 , June 2011 , pp. 317 - 339
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

Footnotes

Part of this work was supported by the Marsden Fund Council from Government funding, administered by the Royal Society of New Zealand.

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