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A criterion of Γ-nullness and differentiability of convex and quasiconvex functions

Jaroslav Tišer, Luděk Zajíček (2015)

Studia Mathematica

We introduce a criterion for a set to be Γ-null. Using it we give a shorter proof of the result that the set of points where a continuous convex function on a separable Asplund space is not Fréchet differentiable is Γ-null. Our criterion also implies a new result about Gâteaux (and Hadamard) differentiability of quasiconvex functions.

A Lipschitz function which is C on a.e. line need not be generically differentiable

Luděk Zajíček (2013)

Colloquium Mathematicae

We construct a Lipschitz function f on X = ℝ ² such that, for each 0 ≠ v ∈ X, the function f is C smooth on a.e. line parallel to v and f is Gâteaux non-differentiable at all points of X except a first category set. Consequently, the same holds if X (with dimX > 1) is an arbitrary Banach space and “a.e.” has any usual “measure sense”. This example gives an answer to a natural question concerning the author’s recent study of linearly essentially smooth functions (which generalize essentially smooth...

A one-sided version of Alexiewicz-Orlicz's differentiability theorem.

E. Corbacho, A. Plichko, V. Tarieladze (2005)

RACSAM

Modificando adecuadamente el método de un trabajo olvidado [1], probamos que si una aplicación continua, de un subconjunto abierto no vacío U de un espacio vectorial topológico metrizable separable y de Baire E, en un espacio localmente convexo, es direccionalmente diferenciable por la derecha en U según un subconjunto comagro de E, entonces, es genéricamente Gâteaux diferenciable en U. Nuestro resultado implica que cualquier espacio vectorial topológico, metrizable, separable y de Baire, es débilmente...

A result on equiabsolute integrability

Cristina Marcelli, Anna Salvadori (1990)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

We prove the equiabsolute integrability of a class of gradients, for functions in W 1 , 1 . The present result appears as the localized version of well-known classical theorems.

Betti's reciprocal theorem for Cosserat elastic shells

Franco Pastrone (1991)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

It is proved that, as in three-dimensional elasticity, Betti's theorem represents a criterion for the existence of a stored-energy function for a Cosserat elastic shell.

Control for the Sine-Gordon equation

Madalina Petcu, Roger Temam (2004)

ESAIM: Control, Optimisation and Calculus of Variations

In this article we apply the optimal and the robust control theory to the sine-Gordon equation. In our case the control is given by the boundary conditions and we work in a finite time horizon. We present at the beginning the optimal control problem and we derive a necessary condition of optimality and we continue by formulating a robust control problem for which existence and uniqueness of solutions are derived.

Control for the sine-gordon equation

Madalina Petcu, Roger Temam (2010)

ESAIM: Control, Optimisation and Calculus of Variations

In this article we apply the optimal and the robust control theory to the sine-Gordon equation. In our case the control is given by the boundary conditions and we work in a finite time horizon. We present at the beginning the optimal control problem and we derive a necessary condition of optimality and we continue by formulating a robust control problem for which existence and uniqueness of solutions are derived.

Decrease of C1,1 property in vector optimization

Dušan Bednařík, Karel Pastor (2009)

RAIRO - Operations Research

In the paper we generalize sufficient and necessary optimality conditions obtained by Ginchev, Guerraggio, Rocca, and by authors with the help of the notion of ℓ-stability for vector functions.

Dense Subdifferentiability and Trustworthiness for Arbitrary Subdifferentials

Jules, Florence, Lassonde, Marc (2010)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: 49J52, 49J50, 58C20, 26B09.We show that the properties of dense subdifferentiability and of trustworthiness are equivalent for any subdifferential satisfying a small set of natural axioms. The proof relies on a remarkable property of the subdifferential of the inf-convolution of two (non necessarily convex) functions. We also show the equivalence of the dense subdifferentiability property with other basic properties of subdifferentials such as a weak* Lipschitz...

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