Bi-Sobolev Solutions to the Prescribed Jacobian Inequality in the Plane\n with L^p Data

2014/08/07 by Julian Fischer, Fischer, Julian, Olivier Kneuss +1
Mathematics · #Analysis of PDEs (math.AP) #Analytic and geometric function theory #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1408.1587

openalex publication_date 2014/08/07 · openalex created_date 2022/09/01 · openalex updated_date 2026/07/28

Abstract

We construct planar bi-Sobolev mappings whose local volume distortion is\nbounded from below by a given function f\∈ L^p with p>1, i.e. bi-Sobolev\nsolutions for the prescribed Jacobian inequality in the plane for right-hand\nsides f\∈ L^p. More precisely, for any 1<q<(p+1)/2 we construct a solution\nwhich belongs to W^1,q and which preserves the boundary pointwise. For\nbounded right-hand sides f\∈ L^\∞, we provide bi-Lipschitz solutions.\nThe basic building block of our construction are Lipschitz maps which stretch a\ngiven compact subset of the unit square by a given factor while preserving the\nboundary. The construction of these stretching maps relies on a slight\nstrengthening of the covering result of Alberti, Cs "ornyei, and Preiss for\nmeasurable planar sets in the case of compact sets.\n

Citations

• Convexity conditions and existence theorems in nonlinear elasticity

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