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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Contraction semigroups for diffusion with drift


Author: R. Seeley
Journal: Trans. Amer. Math. Soc. 283 (1984), 717-728
MSC: Primary 58G32; Secondary 35K15
DOI: https://doi.org/10.1090/S0002-9947-1984-0737895-3
MathSciNet review: 737895
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Abstract: Recently Dodziuk, Karp and Li, and Strichartz have given results on existence and uniqueness of contraction semigroups generated by the Laplacian $ \Delta $ on a manifold $ M$; earlier, Yau gave related results for $ L = \Delta + V$ for a vector field $ V$. The present paper considers $ L = \Delta - V - c$, with $ c$ a real function, and gives conditions for (a) uniqueness of semigroups on the bounded continuous functions, (b) preservation of $ {C_0}$ (functions vanishing at $ \infty $) by the minimal semigroup, and (c) existence and uniqueness of contraction semigroups on $ {L^p}(\mu ),\;1 \leqslant p < \infty $, for an arbitrary smooth density $ \mu $ on $ M$. The conditions concern $ L\rho /\rho $, where $ \rho $ is a smooth function, $ \rho \to \infty $ as $ x \to \infty $. They variously extend, strengthen, and complement the previous results mentioned above.


References [Enhancements On Off] (What's this?)

  • [A] R. Azencott, Behavior of diffusion semigroups at infinity, Bull. Soc. Math. France 102 (1974), 193-240. MR 0356254 (50:8725)
  • [C] E. Calabi, An extension of E. Hopf's maximum principle with an application to geometry, Duke Math. J. 25 (1958), 45-56. MR 0092069 (19:1056e)
  • [CH] R. Courant and D. Hilbert, Methods of mathematical physics, Vol. 2, Interscience, New York, 1962.
  • [D] J. Dodziuk, Maximum principle for parabolic inequalities and the heat flow on open manifolds, Indiana Univ. Math. J. 32 (1983), 703-716. MR 711862 (85e:58140)
  • [K] T. Kato, Schrödinger operators with singular potentials, Israel J. Math. 13 (1972), 135-148. MR 0333833 (48:12155)
  • [KL] Leon Karp and Peter Li, The heat equation on complete Riemannian manifolds (preprint).
  • [S] R. Strichartz, Analysis of the Laplacian on a complete Riemannian manifold, Cornell Univ. (preprint).
  • [Y] S. T. Yau, On the heat kernel of a complete Riemannian manifold, J. Math. Pures Appl. (9) 57 (1978). 191-201. MR 505904 (81b:58041)
  • [Yo] K. Yosida, Functional analysis, Springer, New York, 1974. MR 0350358 (50:2851)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1984-0737895-3
Article copyright: © Copyright 1984 American Mathematical Society

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