Differential Harnack inequalities and Perelman type entropy formulae for subelliptic operators. (English) Zbl 1366.53027
Summary: In this paper, under the generalized curvature-dimension inequality recently introduced by F. Baudoin and N. Garofalo [J. Eur. Math. Soc. (JEMS) 19, No. 1, 151–219 (2017; Zbl 1359.53018)], we obtain differential Harnack inequalities (Theorem 2.1) for the positive solutions to the Schrödinger equation associated to subelliptic operator with potential. As applications of the differential Harnack inequality, we derive the corresponding parabolic Harnack inequality (Theorem 4.1). Also we define the Perelman type entropy associated to subelliptic operators and derive its monotonicity (Theorem 5.3).
MSC:
| 53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |
| 58J35 | Heat and other parabolic equation methods for PDEs on manifolds |
Keywords:
differential Harnack inequalities; Perelman entropy; curvature-dimension inequality; subelliptic operatorsReferences:
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