scholarly journals Łojasiewicz inequality at singular points

2018 ◽  
Vol 147 (3) ◽  
pp. 1109-1117
Author(s):  
Anna Valette
Keyword(s):  
Singular Points ◽  
Łojasiewicz Inequality ◽  
Lojasiewicz Inequality

scholarly journals On the asymptotic behavior of the solutions to parabolic variational inequalities

2020 ◽  
Vol 2020 (768) ◽  
pp. 149-182
Author(s):  
Maria Colombo ◽  
Luca Spolaor ◽  
Bozhidar Velichkov
Keyword(s):  
Asymptotic Behavior ◽  
Variational Inequalities ◽  
Stationary Solution ◽  
Global Solutions ◽  
Singular Points ◽  
Obstacle Problems ◽  
Abstract Result ◽  
Łojasiewicz Inequality ◽  
Lojasiewicz Inequality ◽  
Thin Obstacle

AbstractWe consider various versions of the obstacle and thin-obstacle problems, we interpret them as variational inequalities, with non-smooth constraint, and prove that they satisfy a new constrained Łojasiewicz inequality. The difficulty lies in the fact that, since the constraint is non-analytic, the pioneering method of L. Simon ([22]) does not apply and we have to exploit a better understanding on the constraint itself. We then apply this inequality to two associated problems. First we combine it with an abstract result on parabolic variational inequalities, to prove the convergence at infinity of the strong global solutions to the parabolic obstacle and thin-obstacle problems to a unique stationary solution with a rate. Secondly, we give an abstract proof, based on a parabolic approach, of the epiperimetric inequality, which we then apply to the singular points of the obstacle and thin-obstacle problems.

Łojasiewicz inequality for a pair of semialgebraic functions

2021 ◽  
Vol 166 ◽  
pp. 102927
Author(s):  
Beata Osińska-Ulrych ◽  
Grzegorz Skalski ◽  
Anna Szlachcińska
Keyword(s):  
Łojasiewicz Inequality ◽  
Lojasiewicz Inequality

scholarly journals Proximal Alternating Minimization and Projection Methods for Nonconvex Problems: An Approach Based on the Kurdyka-Łojasiewicz Inequality

2010 ◽  
Vol 35 (2) ◽  
pp. 438-457 ◽  
Author(s):  
Hédy Attouch ◽  
Jérôme Bolte ◽  
Patrick Redont ◽  
Antoine Soubeyran
Keyword(s):  
Projection Methods ◽  
Alternating Minimization ◽  
Nonconvex Problems ◽  
Łojasiewicz Inequality ◽  
Lojasiewicz Inequality

ŁOJASIEWICZ INEQUALITY FOR POLYNOMIAL FUNCTIONS ON NON-COMPACT DOMAINS

2012 ◽  
Vol 23 (04) ◽  
pp. 1250033 ◽  
Author(s):  
DINH SI TIEP ◽  
HA HUY VUI ◽  
NGUYEN THI THAO
Keyword(s):  
Polynomial Functions ◽  
Łojasiewicz Inequality ◽  
Lojasiewicz Inequality

In this paper we give some versions of the Łojasiewicz inequality on non-compact domains for polynomial functions. We also point out some relations between the existence Łojasiewicz inequality and the phenomenon of singularities at infinity.

Łojasiewicz inequality and exponential convergence of the full-range model of CNNs

2010 ◽  
Vol 40 (4) ◽  
pp. 409-419 ◽  
Author(s):  
Mauro Di Marco ◽  
Mauro Forti ◽  
Massimo Grazzini ◽  
Luca Pancioni
Keyword(s):  
Exponential Convergence ◽  
Full Range ◽  
Łojasiewicz Inequality ◽  
Full Range Model ◽  
Lojasiewicz Inequality

THE ŁOJASIEWICZ EXPONENT AT AN EQUILIBRIUM POINT OF A STANDARD CNN IS 1/2

2006 ◽  
Vol 16 (08) ◽  
pp. 2191-2205 ◽  
Author(s):  
MAURO FORTI ◽  
ALBERTO TESI
Keyword(s):  
Vector Field ◽  
Equilibrium Point ◽  
Critical Point ◽  
Vector Fields ◽  
Equilibrium Points ◽  
Łojasiewicz Inequality ◽  
Łojasiewicz Exponent ◽  
Lojasiewicz Inequality ◽  
Output Trajectory ◽  
Lojasiewicz Exponent

In the sixties, Łojasiewicz proved a fundamental inequality for vector fields defined by the gradient of an analytic function, which gives a lower bound on the norm of the gradient in a neighborhood of a (possibly) non-isolated critical point. The inequality involves a number belonging to (0, 1), which depends on the critical point, and is known as the Łojasiewicz exponent. In this paper, a class of vector fields which are defined on a hypercube of ℝn, is considered. Each vector field is the gradient of a quadratic function in the interior of the hypercube, however it is discontinuous on the boundary of the hypercube. An extended Łojasiewicz inequality for this class of vector fields is proved, and it is also shown that the Łojasiewicz exponent at each point where a vector field vanishes is equal to 1/2. The considered fields include a class of vector fields which describe the dynamics of the output trajectories of a standard Cellular Neural Network (CNN) with a symmetric neuron interconnection matrix. By applying the extended Łojasiewicz inequality, it is shown that each output trajectory of a symmetric CNN has finite length, and as a consequence it converges to an equilibrium point. Furthermore, since the Łojasiewicz exponent at each equilibrium point of a symmetric CNN is equal to 1/2, it follows that each (state) trajectory, and each output trajectory, is exponentially convergent toward an equilibrium point, and this is true even in the most general case where the CNN possesses infinitely many nonisolated equilibrium points. In essence, the obtained results mean that standard symmetric CNNs enjoy the property of absolute stability of exponential convergence.

The finiteness property and Łojasiewicz inequality for global semianalytic sets

2005 ◽  
Vol 5 (3) ◽  
Author(s):  
F. Acquistapace ◽  
F. Broglia ◽  
M. Shiota
Keyword(s):  
Finiteness Property ◽  
Łojasiewicz Inequality ◽  
Lojasiewicz Inequality
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