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

2019/05/20 by Goldman, Michael, Merlet, B.
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1905.07905

Abstract

We study a family of non-convex functionals \E\ on the space of measurable functionsu: Ω₁×Ω₂ ⊂ ℝ^n₁×ℝ^n₂ → ℝ. These functionals vanish on the non-convex subset S(Ω₁×Ω₂) formed by functions of the form u(x₁,x₂)=u₁(x₁) or u(x₁,x₂)=u₂(x₂). We investigate under which conditions the converse implication "E(u) = 0 ⇒ u ∈ S(Ω₁×Ω₂)" 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(Ω₁×Ω₂).

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