10.1070/SM2003v194n07ABEH000750
Bogachev, Vladimir I.
Vladimir I.
Bogachev
Röckner, Michael
Michael
Röckner
On L-P-uniqueness of symmetric diffusion operators on Riemannian manifolds
LONDON MATHEMATICAL SOCIETY RUSSIAN ACAD SCIENCES
2003
2010-04-28T12:58:45Z
2019-07-16T13:14:46Z
journal_article
https://pub.uni-bielefeld.de/record/1609835
https://pub.uni-bielefeld.de/record/1609835.json
1064-5616
Let M be a complete Riemannian manifold of dimension d > 1, let P be a measure on M with density exp U with respect to the Riemannian volume, and let Lf = Deltaf + <b, delf>, where U is an element of H-loc(p,1)(M) and b = delU. It is shown that in the case p > d and q is an element of [p', p] the operator L on the domain C-0(infinity)(M) has a unique extension generating a C-0-semigroup on L-q(M, mu), that is, the set (Z - I)(C-0(infinity)(M)) is dense in L-q(M, mu). In particular, the operator L is essentially self-adjoint on L-2(M, mu). A similar result is proved for elliptic operators with non-constant second order part that are formally symmetric with respect to some measure.