Nonsmooth analysis and its applications on Riemannian

manifoldsS. Hosseini

FSDONA 2011, Germany.

Nonsmooth analysis

However, in many aspects of mathematics such as control theory and matrix analysis, problems arise on smooth manifolds!

Unlike Euclidean spaces, a manifold in general does not have a linear structure!

MotivationNonsmooth Functions are often considered on Euclidean spaces!

A useful technique in dealing with the problems on Riemannian manifolds that are local is by using known result in an Euclidean space along the following line:

1. Convert the problem into one in an Euclidean space.

2. Apply corresponding result in an Euclidean space to the problem.

3. Lift the conclusion back onto the manifold.

Therefore, new techniques are needed for dealing with problems on manifolds!

Question; How can we deal with

general problems?Such as the existence of solutions and necessary conditions of optimality for a general problem.

Ekeland variational principle;

Palais -Smale condition;

Our key tools;

Palais-Smale condition,

For smooth functions on Rimannian manifolds

K. C. Chang, For locally Lipschitz functions on Hilbert spaces,1981.

E. Mirenchi, A. Salvatore, For locally Lipschitz

functions on Riemannian manifolds,1994.

K. Le, D. Motreanu, For lower semi continuos functions on Hilbert

spaces,2002.

S. Hosseini, M.R. Pouryayevali, For lower

semi continuous functions on Riemannian

manifolds,2010.

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Definitions

Subderivative of lower semi continuous functions on Riemannian manifolds

Contigent derivative of lower semi continuous functions on Riemannian manifolds

Generalized directional derivative;

Generalized gradient

Generalization of classical derivative

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New Definitions

Bouligand tangent cone

Clarke tangent cone

Clarke normal cone

Palais-Smale condition

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Our main results

Lebourgs Mean Value Theorem

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Applications of nonsmooth analysis on manifolds

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Optimization

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Characterization of epi-Lipschitz subset of Rimannian manifold

Epi-Lipschitz subsets of Rimannian manifolds

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Applications of Epi-Lipschitz subsets

Noncritical neck principal of Morse theory