Sobolev homeomorphisms with gradients of low rank via laminates

2016/03/13 by Faraco, Daniel, Mora-Corral, Carlos, Oliva, Marcos · 1 citation
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1603.04047

Abstract

Let Ω⊂ ℝⁿ be a bounded open set. Given 2≤ m≤ n, we construct a convex function ϕ:Ω→ ℝ whose gradient f= ∇ ϕ is a Hölder continuous homeomorphism, f is the identity on ∂ Ω, the derivative D f has rank m-1 a.e. in Ω and D f is in the weak L^m space L^m,w. The proof is based on convex integration and staircase laminates.

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