We introduce the convex cone constituted by the directions of majoration of a quasiconvex function. This cone is used to formulate a qualification condition ensuring the epiconvergence of a sequence of general quasiconvex marginal functions in finite dimensional spaces.
We construct explicitly piecewise affine mappings u:ℝ ⁿ → ℝ ⁿ with affine boundary data satisfying the constraint div u = 0. As an application of the construction we give short and direct proofs of the main approximation lemmas with constraints in convex integration theory. Our approach provides direct proofs avoiding approximation by smooth mappings and works in all dimensions n ≥ 2. After a slight modification of our construction, the constraint div u = 0 can be turned into det Du = 1, giving...
In this paper we study a model problem describing the movement of a glacier under Glen’s flow law and investigated by Colinge and Rappaz [Colinge and Rappaz, ESAIM: M2AN 33 (1999) 395–406]. We establish error estimates for finite element approximation using the results of Chow [Chow, SIAM J. Numer. Analysis 29 (1992) 769–780] and Liu and Barrett [Liu and Barrett, SIAM J. Numer. Analysis 33 (1996) 98–106] and give an analysis of the convergence of the successive approximations used in [Colinge and...
In this paper we study a model problem describing the movement of a glacier under Glen's flow law and investigated by Colinge and Rappaz [Colinge and Rappaz, ESAIM: M2AN33 (1999) 395–406]. We establish error estimates for finite element approximation using the results of Chow [Chow, SIAM J. Numer. Analysis29 (1992) 769–780] and Liu and Barrett [Liu and Barrett, SIAM J. Numer. Analysis33 (1996) 98–106] and give an analysis of the convergence of the successive approximations used in [Colinge and...
First order characterizations of pseudoconvex functions are investigated in terms of generalized directional derivatives. A connection with the invexity is analysed. Well-known first order characterizations of the solution sets of pseudolinear programs are generalized to the case of pseudoconvex programs. The concepts of pseudoconvexity and invexity do not depend on a single definition of the generalized directional derivative.
2000 Mathematics Subject Classification: 90C46, 90C26, 26B25, 49J52.The constrained optimization problem min f(x), gj(x) ≤ 0 (j = 1,…p) is considered, where f : X → R and gj : X → R are nonsmooth functions with domain X ⊂ Rn. First-order necessary and first-order sufficient optimality conditions are obtained when gj are quasiconvex functions. Two are the main features of the paper: to treat nonsmooth problems it makes use of Dini derivatives; to obtain more sensitive conditions, it admits directionally...
We develop in this paper an improvement of the method given by S. Bobkov and M. Ledoux in [BL00]. Using the Prékopa-Leindler inequality, we prove a modified logarithmic Sobolev inequality adapted for all measures on , with a strictly convex and super-linear potential. This inequality implies modified logarithmic Sobolev inequality, developed in [GGM05, GGM07], for all uniformly strictly convex potential as well as the Euclidean logarithmic Sobolev inequality.
We study a class of functions which differ essentially from those which are the sum of a convex function and a regular one and which have interesting properties related to -convergence and to problems with non-convex constraints. In particular some results are given for the associated evolution equations.
La semiconcavità è una nozione che generalizza quella di concavità conservandone la maggior parte delle proprietà ma permettendo di ampliarne le applicazioni. Questa è una rassegna dei punti più salienti della teoria delle funzioni semiconcave, con particolare riguardo allo studio dei loro insiemi singolari. Come applicazione, si discuterà una formula di rappresentazione per la soluzione di un modello dinamico per la materia granulare.
The author proved in 2018 that if is an open subset of a Hilbert space, continuous functions and a nontrivial modulus such that , is locally semiconvex with modulus and is locally semiconcave with modulus , then there exists such that . This is a generalization of Ilmanen’s lemma (which deals with linear modulus and functions on an open subset of ). Here we extend the mentioned result from Hilbert spaces to some superreflexive spaces, in particular to spaces, . We also prove...