The Lojasiewicz Inequality for Nonsmooth Subanalytic Functions with Applications to Subgradient Dynamical Systems∗

@inproceedings{Bolte2007TheLI,
  title={The Lojasiewicz Inequality for Nonsmooth Subanalytic Functions with Applications to Subgradient Dynamical Systems∗},
  author={J{\'e}r{\^o}me Bolte and Aris Daniilidis and Adrian S. Lewis},
  year={2007}
}
Given a real-analytic function f : Rn → R and a critical point a ∈ Rn, the Lojasiewicz inequality asserts that there exists θ ∈ [ 1 2 , 1) such that the function |f − f(a)|θ ‖∇f‖−1 remains bounded around a. In this paper, we extend the above result to a wide class of nonsmooth functions (that possibly admit the value +∞), by establishing an analogous inequality in which the derivative ∇f(x) can be replaced by any element x∗ of the subdifferential ∂f(x) of f . Like its smooth version, this… CONTINUE READING
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References

Publications referenced by this paper.
Showing 1-10 of 25 references

Proof of the gradient conjecture of R

K. Kurdyka, T. Mostowski, A. Parusiński
Thom, Ann. Math., 152 • 2000
View 5 Excerpts
Highly Influenced

Asymptotics for a class of non-linear evolution equations

L. Simon
with applications to geometric problems, Ann. Math., 118 • 1983
View 5 Excerpts
Highly Influenced

Variational Analysis

R. Tyrrell Rockafellar, Roger J-B Wets
1970
View 5 Excerpts
Highly Influenced

Problèmes rencontrés dans mon parcours mathématique: Un bilan

R. Thom
IHES Publ. Math., 70 • 1989
View 4 Excerpts
Highly Influenced

A Sard theorem for non-differentiable functions

J. Bolte, A. Daniilidis, A. S. Lewis
J. Math. Anal. Appl, 321 • 2006
View 1 Excerpt

A hyperbolic variant of Simon’s convergence theorem

A. Haraux
Evolution Equations and Their Applications in Physical and Life Sciences • 1998
View 1 Excerpt

Asymptotics for a class of non - linear evolution equations , with applications to geometric problems

L. Simon
Ann . Math . • 1998

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