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

J.L. Fischer, O. Kneuss, Journal of Differential Equations 266 (2019) 257–311.

Journal Article | Published | English
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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.
Publishing Year
Date Published
2019-01-05
Journal Title
Journal of Differential Equations
Volume
266
Issue
1
Page
257 - 311
IST-REx-ID

Cite this

Fischer JL, Kneuss O. Bi-Sobolev solutions to the prescribed Jacobian inequality in the plane with L p data and applications to nonlinear elasticity. Journal of Differential Equations. 2019;266(1):257-311. doi:10.1016/j.jde.2018.07.045
Fischer, J. L., & Kneuss, O. (2019). Bi-Sobolev solutions to the prescribed Jacobian inequality in the plane with L p data and applications to nonlinear elasticity. Journal of Differential Equations, 266(1), 257–311. https://doi.org/10.1016/j.jde.2018.07.045
Fischer, Julian L, and Olivier Kneuss. “Bi-Sobolev Solutions to the Prescribed Jacobian Inequality in the Plane with L p Data and Applications to Nonlinear Elasticity.” Journal of Differential Equations 266, no. 1 (2019): 257–311. https://doi.org/10.1016/j.jde.2018.07.045.
J. L. Fischer and O. Kneuss, “Bi-Sobolev solutions to the prescribed Jacobian inequality in the plane with L p data and applications to nonlinear elasticity,” Journal of Differential Equations, vol. 266, no. 1, pp. 257–311, 2019.
Fischer JL, Kneuss O. 2019. Bi-Sobolev solutions to the prescribed Jacobian inequality in the plane with L p data and applications to nonlinear elasticity. Journal of Differential Equations. 266(1), 257–311.
Fischer, Julian L., and Olivier Kneuss. “Bi-Sobolev Solutions to the Prescribed Jacobian Inequality in the Plane with L p Data and Applications to Nonlinear Elasticity.” Journal of Differential Equations, vol. 266, no. 1, Academic Press, 2019, pp. 257–311, doi:10.1016/j.jde.2018.07.045.

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