Characterization of gradient young measures generated by homeomorphisms in the plane

Benešová, Barbora; Kružík, Martin

ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 1, pp. 267-288.

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.

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

J.M. Ball, 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

J.M. Ball and R.D. James, Fine phase mixtures as minimizers of energy. Arch. Ration. Mech. Anal. 100 (1988) 13–52. | DOI | MR | Zbl

B. Benešová, M. Kružík and G. Pathó, Young measures supported on invertible matrices. Appl. Anal. 93 (2014) 105–123. | DOI | MR | Zbl

P.G. Ciarlet, Mathematical Elasticity, Vol. I of Three-dimensional Elasticity. North-Holland, Amsterdam (1988). | MR | Zbl

P.G. Ciarlet and J. Nečas, Injectivity and self-contact in nonlinear elasticity. Arch. Ration. Mech. Anal. 97 (1987) 171–188. | DOI | MR | Zbl

S. Conti and G. Dolzmann, On the theory of relaxation in nonlinear elasticity with constraints on the determinant. Arch. Ration. Mech. Anal. 217 (2015) 413–437. | DOI | MR | Zbl

B. Dacorogna, Direct Methods in the Calculus of Variations. 2nd edn. Springer (2008). | MR | Zbl

S. Daneri and A. Pratelli, A planar bi-Lipschitz extension theorem. Adv. Calc. Var. 8 (2014) 221–266. | MR | Zbl

S. Daneri and A. Pratelli, Smooth approximation of bi-Lipschitz orientation-preserving homeomorphisms. Ann. Inst. Henri Poincaré Anal. Nonlin. 31 (2014) 567–589. | DOI | Numdam | MR | Zbl

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: $L^p$ 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

D. Henao and C. Mora-Corral, 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

T. Iwaniec, L.V. Kovalev and J. Onninen, Diffeomorphic approximation of Sobolev homeomorphisms. Arch. Ration. Mech. Anal. 201 (2011) 1047–1067. | DOI | MR | Zbl

D. Kinderlehrer and P. Pedregal, Characterization of Young measures generated by gradients. Arch. Ration. Mech. Anal. 115 (1991) 329–365. | DOI | MR | Zbl

D. Kinderlehrer and P. Pedregal, Gradient Young measures generated by sequences in Sobolev spaces. J. Geom. Anal. 4 (1994) 59–90. | DOI | MR | Zbl

K. Koumatos, F. Rindler and E. Wiedemann, Orientation-preserving Young measures. Preprint arXiv:1307.1007.v1 (2013). | MR

M. Kružík and M. Luskin, The computation of martensitic microstructure with piecewise laminates. J. Sci. Comput. 19 (2003) 293–308. | DOI | MR | Zbl

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

S. Müller, Q. Tang and B.S. Yan, 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

S. Müller and S.J. Spector, An existence theory for nonlinear elasticity that allows for cavitation. Arch. Ration. Mech. Anal. 131 (1995) 1–66. | DOI | MR | Zbl

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

M.E. Schonbek, 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

L. Tartar, Beyond Young measures. Meccanica 30 (1995) 505–526. | DOI | MR | Zbl

L. Tartar, Mathematical tools for studying oscillations and concentrations: From Young measures to $H$-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

P. Tukia, The planar Schönflies theorem for Lipschitz maps. Ann. Acad. Sci. Fenn. Ser. A I Math. 5 (1980) 49–72. | DOI | MR | Zbl

L.C. Young, 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