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Second‐order operators with degenerate coefficients
Author(s) -
ter Elst A. F. M.,
Robinson Derek W.,
Sikora Adam,
Zhu Yueping
Publication year - 2007
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/pdl017
Subject(s) - mathematics , semigroup , bounded function , ergodic theory , degenerate energy levels , pure mathematics , diagonal , order (exchange) , operator (biology) , hölder condition , contraction (grammar) , mathematical analysis , medicine , biochemistry , chemistry , physics , geometry , finance , repressor , quantum mechanics , transcription factor , economics , gene
Abstract We consider properties of second‐order operators H = − ∑i , j = 1 d∂ ic i j∂ jon ℝ d 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 H 0 and examine properties of the viscosity semigroup S (0) generated by H 0 . The semigroup extends to a positive contraction semigroup on the L p ‐spaces with p ∈ [1, ∞]. We establish that it conserves probability and satisfies L 2 off‐diagonal bounds, and that the wave equation associated with H 0 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 C 2−ε (ℝ d ) with ε > 0.