Second-order operators with degenerate coefficients

A. F M Ter Elst, Derek W. Robinson, Adam Sikora, Yueping Zhu

Research output: Contribution to journalArticlepeer-review

17 Citations (Scopus)

Abstract

We consider properties of second-order operators H = - σ d i,j=1 mathematicl equation presnted on ℝd with bounded real symmetric measurable coefficients. We assume that C = (c ij) ≥ 0 almost everywhere, but allow for the possibility that C is degenerate. We associate with H a canonical self-adjoint viscosity operator HO and examine properties of the viscosity semigroup S(0) generated by HO. The semigroup extends to a positive contraction semigroup on the Lp-spaces with p [1,8]. We establish that it conserves probability and satisfies L2 off-diagonal bounds, and that the wave equation associated with HO has finite speed of propagation. Nevertheless, S(0) is not always strictly positive because separation of the system can occur even for subelliptic operators. This demonstrates that subelliptic semigroups are not ergodic in general and their kernels are neither strictly positive nor Hölder continuous. In particular, one can construct examples for which both upper and lower Gaussian bounds fail even with coefficients in C2-e(ℝd) with e < 0.

Original languageEnglish
Pages (from-to)299-328
Number of pages30
JournalProceedings of the London Mathematical Society
Volume95
Issue number2
DOIs
Publication statusPublished - Sep 2007
Externally publishedYes

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