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A Second Order Smooth Variational Principle on Riemannian Manifolds

Published online by Cambridge University Press:  20 November 2018

Daniel Azagra  and
Robb Fry
Daniel Azagra
Affiliation:
ICMAT(CSIC-UAM-UC3-UCM), Departamento de Análisis Matemático, Facultad Ciencias Matemáticas, Universidad Complutense, 28040 Madrid, Spain, e-mail: azagra@mat.ucm.es
Robb Fry
Affiliation:
Department of Mathematics and Statistics, School of Advanced Technologies and Mathematics, Thompson Rivers University, Kamloops, BC V2C 2N5, e-mail: rfry@tru.ca
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Abstract

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We establish a second order smooth variational principle valid for functions defined on (possibly infinite-dimensional) Riemannian manifolds which are uniformly locally convex and have a strictly positive injectivity radius and bounded sectional curvature.

Keywords

58E3049J5246T0547J3058B20smooth variational principleRiemannian manifold
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 62 , Issue 2 , 01 April 2010 , pp. 241 - 260
Copyright
Copyright © Canadian Mathematical Society 2010

References

[1] Azagra, D. and Cepedello Boiso, M., Uniform approximation of continuous mappings by smooth mappings with no critical points on Hilbert manifolds. Duke Math. J. 124(2004), 47–66. doi:10.1215/S0012-7094-04-12412-1Google Scholar
[2] Azagra, D., Ferrera, J., and F. López-Mesas, Nonsmooth analysis and Hamilton–Jacobi equations on Riemannian manifolds. J. Funct. Anal. 220(2005), 304–361. doi:10.1016/j.jfa.2004.10.008Google Scholar
[3] Azagra, D., Ferrera, J., F. López-Mesas, and Rangel, Y., Smooth approximation of Lipschitz functions on Riemannian manifolds. J. Math. Anal. Appl. 326(2007), 1370–1378. doi:10.1016/j.jmaa.2006.03.088Google Scholar
[4] Azagra, D., Ferrera, J., and Sanz, B., Viscosity solutions to second order partial differential equations on Riemannian manifolds. J. Differential Equations 245(2008), no. 2, 307–336. doi:10.1016/j.jde.2008.03.030Google Scholar
[5] Borwein, J. M. and Preiss, D., A smooth variational principle with applications to subdifferentiability and to differentiability of convex functions. Trans. Amer. Math. Soc. 303(1987), 517–527. doi:10.2307/2000681Google Scholar
[6] Borwein, Jonathan M. and Zhu, Qiji J., Viscosity solutions and viscosity subderivatives in smooth Banach spaces with applications to metric regularity. SIA M J. Control Optim. 34(1996), 1568–1591. doi:10.1137/S0363012994268801Google Scholar
[7] Deville, R., Godefroy, G., and Zizler, V., Smoothness and renormings in Banach spaces. Pitman Monographs and Surveys in Pure and Applied Mathematics 64. Longman, 1993.Google Scholar
[8] Deville, R., A smooth variational principle with applications to Hamilton–Jacobi equations in infinite dimensions. J. Funct. Anal. 111(1993), 197–212. doi:10.1006/jfan.1993.1009Google Scholar
[9] Deville, R. and Ghoussoub, N., Perturbed minimization principles and applications. In: Handbook of the Geometry of Banach Spaces, Vol. I, North-Holland, Amsterdam, 2001, 393–435.Google Scholar
[10] do Carmo, M. P., Riemannian Geometry. Mathematics: Theory and Applications, Birkhäuser Boston, 1992.Google Scholar
[11] Ekeland, I., On the variational principle. J. Math. Anal. Appl. 47(1974), 324–353. doi:10.1016/0022-247X(74)90025-0Google Scholar
[12] Klingenberg, W., Riemannian Geometry. Walter de Gruyter Studies in Mathematics 1. Berlin–New York, 1982.Google Scholar
[13] Lang, S., Fundamentals of Differential Geometry. Graduate Texts in Math. 191. Springer-Verlag, New York, 1999.Google Scholar
[14] Sakai, T., Riemannian Geometry. Transl. Math. Monogr.149. American Mathematical Society, Providence, RI, 1992.Google Scholar
[15] Spivak, M., A Comprehensive Introduction to Differential Geometry, Vol. IV. Second edition. Publish or Perish, Wilmington, DE, 1979.Google Scholar