Bi-Sobolev solutions to the prescribed Jacobian inequality in the plane with L p data and applications to nonlinear elasticity Journal Article


Author(s): Fischer, Julian; Kneuss, Olivier
Article Title: Bi-Sobolev solutions to the prescribed Jacobian inequality in the plane with L p data and applications to nonlinear elasticity
Affiliation IST Austria
Abstract: We construct planar bi-Sobolev mappings whose local volume distortion is bounded from below by a given function f∈Lp with p>1. More precisely, for any 1<q<(p+1)/2 we construct W1,q-bi-Sobolev maps with identity boundary conditions; for f∈L∞, we provide bi-Lipschitz maps. The basic building block of our construction are bi-Lipschitz maps which stretch a given compact subset of the unit square by a given factor while preserving the boundary. The construction of these stretching maps relies on a slight strengthening of the celebrated covering result of Alberti, Csörnyei, and Preiss for measurable planar sets in the case of compact sets. We apply our result to a model functional in nonlinear elasticity, the integrand of which features fast blowup as the Jacobian determinant of the deformation becomes small. For such functionals, the derivation of the equilibrium equations for minimizers requires an additional regularization of test functions, which our maps provide.
Keywords: Nonlinear elasticity; Prescribed Jacobian inequality
Journal Title: Journal of Differential Equations
Volume: 266
Issue 1
ISSN: 00220396
Publisher: Academic Press  
Date Published: 2018-07-30
Start Page: 257
End Page: 311
DOI: 10.1016/j.jde.2018.07.045
Open access: no
IST Austria Authors
  1. Julian Fischer
    17 Fischer
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