Non-convex functionals penalizing simultaneous ...
Document type :
Compte-rendu et recension critique d'ouvrage
Title :
Non-convex functionals penalizing simultaneous oscillations along independent directions: rigidity estimates
Author(s) :
Goldman, Michael [Auteur]
Laboratoire Jacques-Louis Lions [LJLL (UMR_7598)]
Merlet, Benoît [Auteur]
Reliable numerical approximations of dissipative systems [RAPSODI]
Laboratoire Jacques-Louis Lions [LJLL (UMR_7598)]
Merlet, Benoît [Auteur]
Reliable numerical approximations of dissipative systems [RAPSODI]
Journal title :
Annali della Scuola Normale Superiore di Pisa, Classe di Scienze
Pages :
1473--1509
Publisher :
Scuola Normale Superiore
Publication date :
2021
ISSN :
0391-173X
HAL domain(s) :
Mathématiques [math]/Equations aux dérivées partielles [math.AP]
English abstract : [en]
We study a family of non-convex functionals {E} on the space of measurable functions u : Ω 1 × Ω 2 ⊂ R n 1 × R n 2 → R. These functionals vanish on the non-convex subset S(Ω 1 × Ω 2) formed by functions of the form u(x 1 ...
Show more >We study a family of non-convex functionals {E} on the space of measurable functions u : Ω 1 × Ω 2 ⊂ R n 1 × R n 2 → R. These functionals vanish on the non-convex subset S(Ω 1 × Ω 2) formed by functions of the form u(x 1 , x 2) = u 1 (x 1) or u(x 1 , x 2) = u 2 (x 2). We investigate under which conditions the converse implication "E(u) = 0 ⇒ u ∈ S(Ω 1 × Ω 2)" holds. In particular, we show that the answer depends strongly on the smoothness of u. We also obtain quantitative versions of this implication by proving that (at least for some parameters) E(u) controls in a strong sense the distance of u to S(Ω 1 × Ω 2).Show less >
Show more >We study a family of non-convex functionals {E} on the space of measurable functions u : Ω 1 × Ω 2 ⊂ R n 1 × R n 2 → R. These functionals vanish on the non-convex subset S(Ω 1 × Ω 2) formed by functions of the form u(x 1 , x 2) = u 1 (x 1) or u(x 1 , x 2) = u 2 (x 2). We investigate under which conditions the converse implication "E(u) = 0 ⇒ u ∈ S(Ω 1 × Ω 2)" holds. In particular, we show that the answer depends strongly on the smoothness of u. We also obtain quantitative versions of this implication by proving that (at least for some parameters) E(u) controls in a strong sense the distance of u to S(Ω 1 × Ω 2).Show less >
Language :
Anglais
Popular science :
Non
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