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Nonsmooth analysis on smooth manifolds


Authors: Yu. S. Ledyaev and Qiji J. Zhu
Journal: Trans. Amer. Math. Soc. 359 (2007), 3687-3732
MSC (2000): Primary 93D05, 93D20, 34D20
DOI: https://doi.org/10.1090/S0002-9947-07-04075-5
Published electronically: February 23, 2007
MathSciNet review: 2302512
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Abstract: We study infinitesimal properties of nonsmooth (nondifferentiable) functions on smooth manifolds. The eigenvalue function of a matrix on the manifold of symmetric matrices gives a natural example of such a nonsmooth function.

A subdifferential calculus for lower semicontinuous functions is developed here for studying constrained optimization problems, nonclassical problems of calculus of variations, and generalized solutions of first-order partial differential equations on manifolds. We also establish criteria for monotonicity and invariance of functions and sets with respect to solutions of differential inclusions.


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Additional Information

Yu. S. Ledyaev
Affiliation: Department of Mathematics, Western Michigan University, Kalamazoo, Michigan 49008 – and – Steklov Institute of Mathematics, Moscow 117966, Russia
Email: ledyaev@wmich.edu

Qiji J. Zhu
Affiliation: Department of Mathematics, Western Michigan University, Kalamazoo, Michigan 49008
Email: zhu@wmich.edu

DOI: https://doi.org/10.1090/S0002-9947-07-04075-5
Keywords: Differential manifolds, nonsmooth analysis, calculus of semicontinuous functions on manifolds, differential inclusions on manifolds, monotonicity and invariance.
Received by editor(s): June 20, 2003
Received by editor(s) in revised form: May 5, 2005
Published electronically: February 23, 2007
Additional Notes: The first author was supported in part by NSF grant #0102496 and by the Russian Fund for Fundamental Research Grant # 02-01-00769.
The second author was supported in part by NSF grants #9704203, #0102496 and by the Faculty Research and Creative Activities Support Fund at Western Michigan University.
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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