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Journal Articles Annali della Scuola Normale Superiore di Pisa, Classe di Scienze Year : 2021

Non-convex functionals penalizing simultaneous oscillations along independent directions: rigidity estimates

Abstract

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).
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Dates and versions

hal-02132896 , version 1 (17-05-2019)

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Michael Goldman, Benoît Merlet. Non-convex functionals penalizing simultaneous oscillations along independent directions: rigidity estimates. Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, 2021, 22 (3), pp.1473--1509. ⟨10.2422/2036-2145.201906_006⟩. ⟨hal-02132896⟩
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