Proximal smoothness and the lower- property.
Quasi-algebraic representability of sets in .
Quasiconvex relaxation of multidimensional control problems with integrands f(t, ξ, v)
We prove a general relaxation theorem for multidimensional control problems of Dieudonné-Rashevsky type with nonconvex integrands f(t, ξ, v) in presence of a convex control restriction. The relaxed problem, wherein the integrand f has been replaced by its lower semicontinuous quasiconvex envelope with respect to the gradient variable, possesses the same finite minimal value as the original problem, and admits a global minimizer. As an application, we provide existence theorems for the image registration...
Quasiconvex relaxation of multidimensional control problems with integrands f(t, ξ, v)
We prove a general relaxation theorem for multidimensional control problems of Dieudonné-Rashevsky type with nonconvex integrands f(t, ξ, v) in presence of a convex control restriction. The relaxed problem, wherein the integrand f has been replaced by its lower semicontinuous quasiconvex envelope with respect to the gradient variable, possesses the same finite minimal value as the original problem, and admits a global minimizer. As an application, we provide existence theorems for the image registration...
Quelques remarques sur les classes de Baire des fonctions de deux variables
Regular mappings between dimensions
The notions of Lipschitz and bilipschitz mappings provide classes of mappings connected to the geometry of metric spaces in certain ways. A notion between these two is given by regular mappings (reviewed in Section 1), in which some non-bilipschitz behavior is allowed, but with limitations on this, and in a quantitative way. In this paper we look at a class of mappings called (s, t)-regular mappings. These mappings are the same as ordinary regular mappings when s = t, but otherwise they behave somewhat...
Restricted continuity and a theorem of Luzin
Let P(X,ℱ) denote the property: For every function f: X × ℝ → ℝ, if f(x,h(x)) is continuous for every h: X → ℝ from ℱ, then f is continuous. We investigate the assumptions of a theorem of Luzin, which states that P(ℝ,ℱ) holds for X = ℝ and ℱ being the class C(X) of all continuous functions from X to ℝ. The question for which topological spaces P(X,C(X)) holds was investigated by Dalbec. Here, we examine P(ℝⁿ,ℱ) for different families ℱ. In particular, we notice that P(ℝⁿ,"C¹") holds, where...
Restrictions of smooth functions to a closed subset
We first provide an approach to the conjecture of Bierstone-Milman-Pawłucki on Whitney’s problem on extendability of functions. For example, the conjecture is affirmative for classical fractal sets. Next, we give a sharpened form of Spallek’s theorem on flatness.
Second-order sufficient condition for -stable functions
The aim of our article is to present a proof of the existence of local minimizer in the classical optimality problem without constraints under weaker assumptions in comparisons with common statements of the result. In addition we will provide rather elementary and self-contained proof of that result.
Semidifferentials, quadratic forms and fully nonlinear elliptic equations of second order
Separate approximate continuity and strong approximate continuity
Separately continuous functions: approximations, extensions, and restrictions.
Solution to the gradient problem of C.E. Weil.
In this paper we give a complete answer to the famous gradient problem of C. E. Weil. On an open set G ⊂ R2 we construct a differentiable function f: G → R for which there exists an open set Ω1 ⊂ R2 such that ∇f(p) ∈ Ω1 for a p ∈ G but ∇f(q) ∉ Ω1 for almost every q ∈ G. This shows that the Denjoy-Clarkson property does not hold in higher dimensions.
Some properties of real functions of two variables and some consequences
Some results on (-pre, )-continuous functions.
Some sufficient conditions for regular approximate differentiability
Sommes de carrés de fonctions dérivables
On montre que toute fonction positive de classe définie sur un intervalle de est somme de deux carrés de fonctions de classe . En dimension 2, toute fonction positive de classe est somme d’un nombre fini de carrés de fonctions de classe , pourvu que ses dérivées d’ordre 4 s’annulent aux points où et s’annulent.
Structural properties of singularities of semiconcave functions
Subanalytic version of Whitney's extension theorem
For any subanalytic -Whitney field (k finite), we construct its subanalytic -extension to . Our method also applies to other o-minimal structures; e.g., to semialgebraic Whitney fields.
Sur les classes de Baire des fonctions de deux variables