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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
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.

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  • [A] Robert Azencott, Behavior of diffusion semi-groups at infinity, Bull. Soc. Math. France 102 (1974), 193–240. MR 0356254
  • [C] E. Calabi, An extension of E. Hopf’s maximum principle with an application to Riemannian geometry, Duke Math. J. 25 (1958), 45–56. MR 0092069
  • [CH] R. Courant and D. Hilbert, Methods of mathematical physics, Vol. 2, Interscience, New York, 1962.
  • [D] Jozef Dodziuk, Maximum principle for parabolic inequalities and the heat flow on open manifolds, Indiana Univ. Math. J. 32 (1983), no. 5, 703–716. MR 711862,
  • [K] Tosio Kato, Schrödinger operators with singular potentials, Proceedings of the International Symposium on Partial Differential Equations and the Geometry of Normed Linear Spaces (Jerusalem, 1972), 1972, pp. 135–148 (1973). MR 0333833,
  • [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] Shing Tung Yau, On the heat kernel of a complete Riemannian manifold, J. Math. Pures Appl. (9) 57 (1978), no. 2, 191–201. MR 505904
  • [Yo] Kôsaku Yosida, Functional analysis, 4th ed., Springer-Verlag, New York-Heidelberg, 1974. Die Grundlehren der mathematischen Wissenschaften, Band 123. MR 0350358

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Article copyright: © Copyright 1984 American Mathematical Society