Recent results on non-convex functionals penalizing oblique oscillations

Goldman, Michael; Merlet, Benoît

Type de document :

Article dans une revue scientifique: Article original

Titre :

Recent results on non-convex functionals penalizing oblique oscillations

Auteur(s) :

Goldman, Michael [Auteur]
Laboratoire Jacques-Louis Lions [LJLL]
Merlet, Benoît [Auteur]

Reliable numerical approximations of dissipative systems [RAPSODI]

Titre de la revue :

Rendiconti del Seminario Matematico

Éditeur :

Università degli studi di Torino / Politecnico di Torino

Date de publication :

2019

ISSN :

0373-1243

Discipline(s) HAL :

Mathématiques [math]/Equations aux dérivées partielles [math.AP]

Résumé en anglais : [en]

The aim of this note is to review some recent results on a family of functionals penalizing oblique oscillations. These functionals naturally appeared in some varia-tional problem related to pattern formation and are ...
Lire la suite >The aim of this note is to review some recent results on a family of functionals penalizing oblique oscillations. These functionals naturally appeared in some varia-tional problem related to pattern formation and are somewhat reminiscent of those introduced by Bourgain, Brezis and Mironescu to characterize Sobolev functions. We obtain both qualitative and quantitative results for functions of finite energy. It turns out that this problem naturally leads to the study of various differential inclusions and has connections with branched transportation models. We review in this paper some recent results obtained in [GM19a, GM19b, GM19c] on non-convex functionals penalizing oblique oscillations. We are mainly interested in both qualitative and quantitative rigidity results for functions with finite energy. We also obtain concentration and rectifiability properties of the corresponding 'defect' measures. We will focus here on the most important results and sacrifice generality for clarity. In particular, we will restrict ourselves to a periodic setting to avoid boundary effects. 1 The energy For n 1 , n 2 ≥ 1 and n = n 1 + n 2 , we decompose R n = X 1 ⊕ X 2 with X 1 ⊥ X 2 and n l = dim X l and consider the n dimensional torus T n := (R/Z) n = T n 1 ⊕ T n 2. For x ∈ R n we write x = x 1 + x 2 its decomposition in X 1 ⊕ X 2. For (x, z) ∈ T n × R n , we introduce the notation Du(x, z) := u(x + z) − u(x) for the discrete derivative. We also fix a radial non-negative kernel 1 ρ ∈ L 1 (R n , R +) with R n ρdx = 1, and supp ρ ⊂ B 1 .Lire moins >

Langue :

Anglais

Comité de lecture :

Oui

Audience :

Internationale

Vulgarisation :

Non

Collections :

Laboratoire Paul Painlevé - UMR 8524

Source :

Harvested from HAL

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