We characterize Young measures generated by gradients of bi-Lipschitz orientation-preserving maps in the plane. This question is motivated by variational problems in nonlinear elasticity where the orientation preservation and injectivity of the admissible deformations are key requirements. These results enable us to derive new weak lower semicontinuity results for integral functionals depending on gradients. As an application, we show the existence of a minimizer for an integral functional with nonpolyconvex energy density among bi-Lipschitz homeomorphisms.
DOI: 10.1051/cocv/2015003
Mots-clés : Orientation-preserving mappings, Young measures
@article{COCV_2016__22_1_267_0, author = {Bene\v{s}ov\'a, Barbora and Kru\v{z}{\'\i}k, Martin}, title = {Characterization of gradient young measures generated by homeomorphisms in the plane}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {267--288}, publisher = {EDP-Sciences}, volume = {22}, number = {1}, year = {2016}, doi = {10.1051/cocv/2015003}, zbl = {1335.49023}, mrnumber = {3489385}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2015003/} }
TY - JOUR AU - Benešová, Barbora AU - Kružík, Martin TI - Characterization of gradient young measures generated by homeomorphisms in the plane JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2016 SP - 267 EP - 288 VL - 22 IS - 1 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2015003/ DO - 10.1051/cocv/2015003 LA - en ID - COCV_2016__22_1_267_0 ER -
%0 Journal Article %A Benešová, Barbora %A Kružík, Martin %T Characterization of gradient young measures generated by homeomorphisms in the plane %J ESAIM: Control, Optimisation and Calculus of Variations %D 2016 %P 267-288 %V 22 %N 1 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2015003/ %R 10.1051/cocv/2015003 %G en %F COCV_2016__22_1_267_0
Benešová, Barbora; Kružík, Martin. Characterization of gradient young measures generated by homeomorphisms in the plane. ESAIM: Control, Optimisation and Calculus of Variations, Volume 22 (2016) no. 1, pp. 267-288. doi : 10.1051/cocv/2015003. http://archive.numdam.org/articles/10.1051/cocv/2015003/
R.A. Adams and J.J.F. Fournier, Sobolev spaces, 2nd edn. Elsevier, Amsterdam (2003). | MR | Zbl
K. Astala and D. Faraco, Quasiregular mappings and Young measures. In vol. 132. Proc. of Royal Soc. Edinb. A (2002) 1045–1056. | MR | Zbl
Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal. 63 (1977) 337–403. | DOI | MR | Zbl
,J.M. Ball, Global invertibility of Sobolev functions and the interpenetration of matter. In vol. 88. Proc. of Roy. Soc. Edinb. A (1981) 315–328. | MR | Zbl
J.M. Ball, A version of the fundamental theorem for Young measures. In: PDEs and Continuum Models of Phase Transition, edited by M. Rascle, D. Serre, M. Slemrod. Vol. 344 of Lect. Notes Phys. Springer, Berlin (1989) 207–215. | MR | Zbl
J.M. Ball, Some open problems in elasticity. In Geometry, Mechanics, and Dynamics. Springer, New York (2002) 3–59. | MR | Zbl
Fine phase mixtures as minimizers of energy. Arch. Ration. Mech. Anal. 100 (1988) 13–52. | DOI | MR | Zbl
and ,Young measures supported on invertible matrices. Appl. Anal. 93 (2014) 105–123. | DOI | MR | Zbl
, and ,P.G. Ciarlet, Mathematical Elasticity, Vol. I of Three-dimensional Elasticity. North-Holland, Amsterdam (1988). | MR | Zbl
Injectivity and self-contact in nonlinear elasticity. Arch. Ration. Mech. Anal. 97 (1987) 171–188. | DOI | MR | Zbl
and ,On the theory of relaxation in nonlinear elasticity with constraints on the determinant. Arch. Ration. Mech. Anal. 217 (2015) 413–437. | DOI | MR | Zbl
and ,B. Dacorogna, Direct Methods in the Calculus of Variations. 2nd edn. Springer (2008). | MR | Zbl
A planar bi-Lipschitz extension theorem. Adv. Calc. Var. 8 (2014) 221–266. | MR | Zbl
and ,Smooth approximation of bi-Lipschitz orientation-preserving homeomorphisms. Ann. Inst. Henri Poincaré Anal. Nonlin. 31 (2014) 567–589. | DOI | Numdam | MR | Zbl
and ,I. Fonseca and W. Gangbo, Degree Theory in Analysis and Applications. Clarendon Press, Oxford (1995). | MR | Zbl
L.C. Evans and R.F. Gariepy, Measure Theory and Fine Properties of Functions. CRC Press, Inc. Boca Raton (1992). | MR | Zbl
I. Fonseca and G. Leoni, Modern Methods in the Calculus of Variations: Spaces. Springer (2007). | MR | Zbl
M. Giaquinta, G. Modica and J. Souček, Cartesian currents in the calculus of variations. Vol. I and II. Springer (1998). | Numdam | MR | Zbl
Invertibility and weak continuity of the determinant for the modelling of cavitation and fracture in nonlinear elasticity. Arch. Ration. Mech. Anal. 197 (2010) 619–655. | DOI | MR | Zbl
and ,Diffeomorphic approximation of Sobolev homeomorphisms. Arch. Ration. Mech. Anal. 201 (2011) 1047–1067. | DOI | MR | Zbl
, and ,Characterization of Young measures generated by gradients. Arch. Ration. Mech. Anal. 115 (1991) 329–365. | DOI | MR | Zbl
and ,Gradient Young measures generated by sequences in Sobolev spaces. J. Geom. Anal. 4 (1994) 59–90. | DOI | MR | Zbl
and ,K. Koumatos, F. Rindler and E. Wiedemann, Orientation-preserving Young measures. Preprint arXiv:1307.1007.v1 (2013). | MR
The computation of martensitic microstructure with piecewise laminates. J. Sci. Comput. 19 (2003) 293–308. | DOI | MR | Zbl
and ,C.B. Morrey, Multiple Integrals in the Calculus of Variations. Springer, Berlin (1966). | MR | Zbl
S. Müller, Variational models for microstructure and phase transisions. Vol. 1713 of Lect. Notes Math. Springer Berlin (1999) 85–210. | MR | Zbl
On a new class of elastic deformations not allowing for cavitation. Ann. Inst. Henri Poincaré Anal. Nonlin. 11 (1994) 217–243. | DOI | Numdam | MR | Zbl
, and ,An existence theory for nonlinear elasticity that allows for cavitation. Arch. Ration. Mech. Anal. 131 (1995) 1–66. | DOI | MR | Zbl
and ,P. Pedregal, Parametrized Measures and Variational Principles. Birkäuser, Basel (1997). | MR | Zbl
T. Roubíček, Relaxation in Optimization Theory and Variational Calculus. W. de Gruyter, Berlin (1997). | MR | Zbl
Convergence of solutions to nonlinear dispersive equations. Commun. Partial Differ. Equ. 7 (1982) 959–1000. | DOI | MR | Zbl
,Q. Tang, Almost-everywhere injectivity in nonlinear elasticity. In vol. 109. Proc. Roy. Soc. Edinb. A (1988) 79–95. | MR | Zbl
Beyond Young measures. Meccanica 30 (1995) 505–526. | DOI | MR | Zbl
,L. Tartar, Mathematical tools for studying oscillations and concentrations: From Young measures to -measures and their variants. Multiscale problems in science and technology. Challenges to mathematical analysis and perspectives. Edited by N. Antoničet al. Proc. of the conference on multiscale problems in science and technology, held in Dubrovnik, Croatia, September 3-9, 2000. Springer, Berlin (2002). | MR | Zbl
The planar Schönflies theorem for Lipschitz maps. Ann. Acad. Sci. Fenn. Ser. A I Math. 5 (1980) 49–72. | DOI | MR | Zbl
,Generalized curves and the existence of an attained absolute minimum in the calculus of variations. C. R. Soc. Sci. Lett. Varsovie, Classe III 30 (1937) 212–234. | JFM | Zbl
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