Abstract
This work establishes a characterization theorem for (generalized) Young measures generated by symmetric derivatives of functions of bounded deformation (BD) in the spirit of the classical Kinderlehrer–Pedregal theorem. Our result places such Young measures in duality with symmetric-quasiconvex functions with linear growth. The “local” proof strategy combines blow-up arguments with the singular structure theorem in BD (the analogue of Alberti’s rank-one theorem in BV), which was recently proved by the authors. As an application of our characterization theorem we show how an atomic part in a BD-Young measure can be split off in generating sequences.
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Alberti G.: Rank one property for derivatives of functions with bounded variation. Proc. R. Soc. Edinburgh Sect. A 123, 239–274 (1993)
Alibert J.J., Bouchitté G.: Non-uniform integrability and generalized Young measures. J. Convex Anal. 4, 129–147 (1997)
Ambrosio L., Coscia A., Dal Maso G.: Fine properties of functions with bounded deformation. Arch. Ration. Mech. Anal. 139, 201–238 (1997)
Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free-Discontinuity Problems, Oxford Mathematical Monographs, Oxford University Press, 2000
Arroyo-Rabasa, A., De Philippis, G., Rindler, F.: Lower semicontinuity and relaxation of linear-growth integral functionals under PDE constraints, arXiv:1701.02230
Babadjian J.-F.: Traces of functions of bounded deformation. Indiana Univ. Math. J. 64, 1271–1290 (2015)
Baia M., Matias J., Santos P.: Characterization of generalized Young measures in the \({\mathcal{A}}\)-quasiconvexity context. Indiana Univ. Math. J. 62, 487–521 (2013)
Ball, J.M.: A version of the fundamental theorem for Young measures, PDEs and continuum models of phase transitions (Nice, 1988), Lecture Notes in Physics, vol. 344, Springer, pp. 207–215, 1989
Ball J.M., James R.D.: Fine phase mixtures as minimizers of energy. Arch. Ration. Mech. Anal. 100, 13–52 (1987)
Barroso A.C., Fonseca I., Toader R.: A relaxation theorem in the space of functions of bounded deformation. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 29, 19–49 (2000)
Benešová, B., Kružík, M.: Characterization of gradient Young measures generated by homeomorphisms in the plane. ESAIM Control Optim. Calc. Var. 22, 267–288 (2016)
Conti S., Faraco D., Maggi F.: A new approach to counterexamples to L 1 estimates: Korn’s inequality, geometric rigidity, and regularity for gradients of separately convex functions. Arch. Ration. Mech. Anal. 175, 287–300 (2005)
Conti, S., Focardi, M., Iurlano, F.: Which special functions of bounded deformation have bounded variation? Proc. Roy. Soc. Edinburgh Sect. A. arXiv:1502.07464 (2015)
Conway, J.B.: A Course in Functional Analysis, 2nd ed., Graduate Texts in Mathematics, vol. 96, Springer, 1990
Dacorogna, B.: Direct Methods in the Calculus of Variations, 2nd ed., Applied Mathematical Sciences, vol. 78, Springer, 2008
De Philippis, G., Rindler, F.: On the structure of \({\mathcal{A}}\)-free measures and applications. Ann. Math. 184, 1017–1039 (2016)
DiPerna R.J., Majda A.J.: Oscillations and concentrations in weak solutions of the incompressible fluid equations. Comm. Math. Phys. 108, 667–689 (1987)
Ebobisse F.: On lower semicontinuity of integral functionals in \({LD(\Omega)}\). Ricerche Mat. 49, 65–76 (2000)
Fonseca I., Kružík M.: Oscillations and concentrations generated by \({\mathcal{A}}\)-free mappings and weak lower semicontinuity of integral functionals. ESAIM Control Optim. Calc. Var. 16, 472–502 (2010)
Fonseca I., Müller S.: \({\mathcal{A}}\)-quasiconvexity, lower semicontinuity, and Young measures. SIAM J. Math. Anal. 30, 1355–1390 (1999)
Fonseca I., Müller S., Pedregal P.: Analysis of concentration and oscillation effects generated by gradients. SIAM J. Math. Anal. 29, 736–756 (1998)
Kałamajska A., Kružík M.: Oscillations and concentrations in sequences of gradients. ESAIM Control Optim. Calc. Var. 14, 71–104 (2008)
Kinderlehrer D., Pedregal P.: Characterizations of Young measures generated by gradients. Arch. Ration. Mech. Anal. 115, 329–365 (1991)
Kinderlehrer D., Pedregal P.: Gradient Young measures generated by sequences in Sobolev spaces. J. Geom. Anal. 4, 59–90 (1994)
Kirchheim B., Kristensen J.: On rank-one convex functions that are homogeneous of degree one. Arch. Ration. Mech. Anal. 221, 527–558 (2016)
Kohn R. V.: New integral estimates for deformations in terms of their nonlinear strains. Arch. Ration. Mech. Anal. 78, 131–172 (1982)
Kristensen J.: Lower semicontinuity in spaces of weakly differentiable functions. Math. Ann. 313, 653–710 (1999)
Kristensen, J., Rindler, F.: Characterization of generalized gradient Young measures generated by sequences in \({{{\rm W}^{1,1}}}\) and BV. Arch. Ration. Mech. Anal. 197, 539–598 (2010), Erratum: Vol. 203, 693–700 (2012)
Kružík, M., Roubíček, T.: Explicit characterization of L p-Young measures. J. Math. Anal. Appl. 198, 830–843 (1996)
Kružík M., Roubíček T.: On the measures of DiPerna and Majda. Math. Bohem. 122, 383–399 (1997)
Massaccesi, A., Vittone, D.: An elementary proof of the rank one theorem for BV functions, To appear on JEMS, 2016
Matthies, H., Strang, G., Christiansen, E.: The saddle point of a differential program, Energy methods in finite element analysis, Wiley, pp. 309–318, 1979
Mattila, P.: Geometry of Sets and Measures in Euclidean Spaces, Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, 1995
Morrey C.B. Jr.: Quasiconvexity and the semicontinuity of multiple integrals. Pacif. J. Math. 2, 25–53 (1952)
Ornstein D.: A non-inequality for differential operators in the L 1 norm. Arch. Ration. Mech. Anal. 11, 40–49 (1962)
Pedregal, P.: Parametrized Measures and Variational Principles, Progress in Nonlinear Differential Equations and their Applications, vol. 30, Birkhäuser, 1997
Rindler, F.: Lower Semicontinuity and Young Measures for Integral Functionals with Linear Growth, Ph.D. thesis, University of Oxford, 2011
Rindler F.: Lower semicontinuity for integral functionals in the space of functions of bounded deformation via rigidity and Young measures. Arch. Ration. Mech. Anal. 202, 63–113 (2011)
Rindler F.: Lower semicontinuity and Young measures in BV without Alberti’s Rank-One Theorem. Adv. Calc. Var. 5, 127–159 (2012)
Rindler F.: A local proof for the characterization of Young measures generated by sequences in BV. J. Funct. Anal. 266, 6335–6371 (2014)
Suquet P.-M.: Existence et régularité des solutions des équations de la plasticité. C. R. Acad. Sci. Paris Sér. A 286, 1201–1204 (1978)
Suquet P.-M.: Un espace fonctionnel pour les équations de la plasticité. Ann. Fac. Sci. Toulouse Math. 1, 77–87 (1979)
Sychev M.A.: A new approach to Young measure theory, relaxation and convergence in energy. Ann. Inst. H. Poincaré Anal. Non Linéaire 16, 773–812 (1999)
Tartar, L.: Compensated compactness and applications to partial differential equations, Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. IV, Res. Notes in Math., vol. 39, Pitman, pp. 136–212, 1979
Tartar, L.: The compensated compactness method applied to systems of conservation laws, Systems of nonlinear partial differential equations (Oxford, 1982), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 111, Reidel, pp. 263–285, 1983
Temam, R.: Mathematical Problems in Plasticity, Gauthier-Villars, 1985
Temam R., Strang G.: Functions of bounded deformation. Arch. Ration. Mech. Anal. 75, 7–21 (1980)
Young L. C.: Generalized curves and the existence of an attained absolute minimum in the calculus of variations. C. R. Soc. Sci. Lett. Varsovie, Cl. III 30, 212–234 (1937)
Young L. C.: Generalized surfaces in the calculus of variations. Ann. of Math. 43, 84–103 (1942)
Young L. C.: Generalized surfaces in the calculus of variations. II. Ann. of Math. 43, 530–544 (1942)
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De Philippis, G., Rindler, F. Characterization of Generalized Young Measures Generated by Symmetric Gradients. Arch Rational Mech Anal 224, 1087–1125 (2017). https://doi.org/10.1007/s00205-017-1096-1
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DOI: https://doi.org/10.1007/s00205-017-1096-1