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On the structure of \({\mathcal A}\)-free measures and applications. (English) Zbl 1352.49050
Consider a \(k\)-th order linear constant-coefficients PDE operator \(\mathcal A\), \((k\in\mathbb N)\), i.e. \(\mathcal A=\sum_{|\alpha|\leq k}A_\alpha\partial^\alpha\), where \(A_\alpha\in\mathbb R^{n\times m}\) and \(\alpha=(\alpha_1,\dots\alpha_d)\) is a multi-index. Assume that \(\mu\) is a finite Radon measure on an open set \(\Omega\subset\mathbb R^d\) with values in \(\mathbb R^m\) and it is \(\mathcal A\)-free. Define the wave cone \(\Lambda_{\mathcal A}=\bigcup_{|\xi|=1}\ker \mathcal A^k(\xi)\subset\mathbb R^m\) with \(\mathcal A^k(\xi)=(2\pi i)^k\sum _{|\alpha |=k}A_\alpha\xi^\alpha\), where \(\xi^\alpha =\xi_1^{\alpha_1}\dots \xi_d^{\alpha_d}\). The authors show that for a measure \(\mu\) solving the initial PDE \(\mathcal A\mu=0\) , the polar \(d\mu/d|\mu|\), i.e. the Radon-Nikodým derivative of \(\mu\) with respect to its total variation measure \(|\mu|\) lies in the wave cone at almost all singular points.
Theorem 1.1. Let \(\Omega \subset \mathbb R^d\) be an open set, let \(\mathcal A\) be a \(k\)-th order linear constant-coefficients differential operator and let \(\mu\in\mathcal M(\Omega;\mathbb R^m)\) be an \(\mathcal A\)-free Radon measure on \(\Omega\) with values in \(\mathbb R^m\). Then, \(d\mu/d|\mu|(x)\in \Lambda_{\mathcal A}\), for \(|\mu|^s\)-a.e. \(x\in\Omega\).
By applying Theorem 1.1. to suitably chosen differential operators, one can easily obtain several remarkable consequences. First of all, a simple proof of Alberti’s rank-one theorem and its extensions to functions of bounded deformation are obtained. Then, the authors prove a structure theorem for the singular part of a finite family of normal currents in the spirit of the rank-one theorem. Considering the results of Alberti and Marquese and of Schioppa, it follows that the Rademacher theorem can hold only for absolutely continuous measures and that every top-dimensional Ambrosio-Kirchheim metric current in \(\mathbb R^d\) is a Federer flat chain.

49Q15 Geometric measure and integration theory, integral and normal currents in optimization
35D30 Weak solutions to PDEs
28B05 Vector-valued set functions, measures and integrals
49Q20 Variational problems in a geometric measure-theoretic setting
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[1] G. Alberti, ”Rank one property for derivatives of functions with bounded variation,” Proc. Roy. Soc. Edinburgh Sect. A, vol. 123, iss. 2, pp. 239-274, 1993. · Zbl 0791.26008
[2] G. Alberti, M. Csörnyei, and D. Preiss, ”Structure of null sets in the plane and applications,” in European Congress of Mathematics, Eur. Math. Soc., Zürich, 2005, pp. 3-22. · Zbl 1088.28002
[3] G. Alberti, M. Csörnyei, and D. Preiss, ”Differentiability of Lipschitz functions, structure of null sets, and other problems,” in Proceedings of the International Congress of Mathematicians. Volume III, 2010, pp. 1379-1394. · Zbl 1251.26010
[4] G. Alberti and A. Marchese, ”On the differentiability of Lipschitz functions with respect to measures in the Euclidean space,” Geom. Funct. Anal., vol. 26, pp. 1-66, 2016. · Zbl 1345.26018
[5] W. K. Allard, ”An integrality theorem and a regularity theorem for surfaces whose first variation with respect to a parametric elliptic integrand is controlled,” in Geometric Measure Theory and the Calculus of Variations, RI: Amer. Math. Soc., Providence, 1986, vol. 44, pp. 1-28. · Zbl 0609.49028
[6] L. Ambrosio, ”Transport equation and Cauchy problem for \(BV\) vector fields,” Invent. Math., vol. 158, iss. 2, pp. 227-260, 2004. · Zbl 1075.35087
[7] L. Ambrosio, ”Transport equation and Cauchy problem for non-smooth vector fields,” in Calculus of Variations and Nonlinear Partial Differential Equations, New York: Springer-Verlag, 2008, vol. 1927, pp. 1-41. · Zbl 1159.35041
[8] L. Ambrosio, A. Coscia, and G. Dal Maso, ”Fine properties of functions with bounded deformation,” Arch. Rational Mech. Anal., vol. 139, iss. 3, pp. 201-238, 1997. · Zbl 0890.49019
[9] L. Ambrosio and G. Dal Maso, ”On the relaxation in \({ BV}(\Omega;{\mathbf R}^m)\) of quasi-convex integrals,” J. Funct. Anal., vol. 109, iss. 1, pp. 76-97, 1992. · Zbl 0769.49009
[10] E. De Giorgi and L. Ambrosio, ”New functionals in the calculus of variations,” Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., vol. 82, iss. 2, pp. 199-210 (1989), 1988. · Zbl 0715.49014
[11] L. Ambrosio, N. Fusco, and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, New York: The Clarendon Press, Oxford University Press, 2000. · Zbl 0957.49001
[12] L. Ambrosio and B. Kirchheim, ”Currents in metric spaces,” Acta Math., vol. 185, iss. 1, pp. 1-80, 2000. · Zbl 0984.49025
[13] G. Bouchitté, I. Fonseca, and L. Mascarenhas, ”A global method for relaxation,” Arch. Rational Mech. Anal., vol. 145, iss. 1, pp. 51-98, 1998. · Zbl 0921.49004
[14] S. Conti and M. Ortiz, ”Dislocation microstructures and the effective behavior of single crystals,” Arch. Ration. Mech. Anal., vol. 176, iss. 1, pp. 103-147, 2005. · Zbl 1064.74144
[15] C. De Lellis, ”A note on Alberti’s rank-one theorem,” in Transport Equations and Multi-D Hyperbolic Conservation Laws, New York: Springer-Verlag, 2008, vol. 5, pp. 61-74. · Zbl 1221.46030
[16] R. J. DiPerna, ”Compensated compactness and general systems of conservation laws,” Trans. Amer. Math. Soc., vol. 292, iss. 2, pp. 383-420, 1985. · Zbl 0606.35052
[17] H. Federer, Geometric Measure Theory, New York: Springer-Verlag, 1969, vol. 153. · Zbl 0176.00801
[18] I. Fonseca and S. Müller, ”Relaxation of quasiconvex functionals in \({ BV}(\Omega,{\mathbf R}^p)\) for integrands \(f(x,u,\nabla u)\),” Arch. Rational Mech. Anal., vol. 123, iss. 1, pp. 1-49, 1993. · Zbl 0788.49039
[19] I. Fonseca and S. Müller, ”\(\mathcalA\)-quasiconvexity, lower semicontinuity, and Young measures,” SIAM J. Math. Anal., vol. 30, iss. 6, pp. 1355-1390, 1999. · Zbl 0940.49014
[20] M. Fuchs and G. Seregin, Variational Methods for Poblems from Pasticity Theory and for Generalized Newtonian Fluids, New York: Springer-Verlag, 2000, vol. 1749. · Zbl 0964.76003
[21] L. Grafakos, Classical Fourier Analysis, Third ed., New York: Springer-Verlag, 2014, vol. 249. · Zbl 1304.42001
[22] L. Grafakos, Modern Fourier Analysis, Third ed., New York: Springer-Verlag, 2014, vol. 250. · Zbl 1304.42002
[23] P. Jones, Product formulas for measures and applications to analysis and geometry.
[24] B. Kirchheim and J. Kristensen, ”On rank one convex functions that are homogeneous of degree one,” Arch. Ration. Mech. Anal., vol. 221, iss. 1, pp. 527-558, 2016. · Zbl 1342.49015
[25] J. Kristensen and F. Rindler, ”Characterization of generalized gradient Young measures generated by sequences in \(W^{1,1}\) and BV,” Arch. Ration. Mech. Anal., vol. 197, iss. 2, pp. 539-598, 2010. · Zbl 1245.49060
[26] J. Kristensen and F. Rindler, ”Relaxation of signed integral functionals in BV,” Calc. Var. Partial Differential Equations, vol. 37, iss. 1-2, pp. 29-62, 2010. · Zbl 1189.49018
[27] J. Kristensen and F. Rindler, ”Piecewise affine approximations for functions of bounded variation,” Numer. Math., vol. 132, pp. 329-346, 2016. · Zbl 1336.41004
[28] A. Massaccesi and D. Vittone, An elementary proof of the rank one theorem for BV functions, 2016.
[29] F. Murat, ”Compacité par compensation,” Ann. Scuola Norm. Sup. Pisa Cl. Sci., vol. 5, iss. 3, pp. 489-507, 1978. · Zbl 0399.46022
[30] F. Murat, ”Compacité par compensation. II,” in Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis, Bologna, 1979, pp. 245-256.
[31] F. Murat, ”Compacité par compensation: condition nécessaire et suffisante de continuité faible sous une hypothèse de rang constant,” Ann. Scuola Norm. Sup. Pisa Cl. Sci., vol. 8, iss. 1, pp. 69-102, 1981. · Zbl 0464.46034
[32] T. O’Neil, ”A measure with a large set of tangent measures,” Proc. Amer. Math. Soc., vol. 123, iss. 7, pp. 2217-2220, 1995. · Zbl 0827.28002
[33] D. Preiss, ”Geometry of measures in \({\mathbf R}^n\): distribution, rectifiability, and densities,” Ann. of Math., vol. 125, iss. 3, pp. 537-643, 1987. · Zbl 0627.28008
[34] D. Preiss, ”Differentiability of Lipschitz functions on Banach spaces,” J. Funct. Anal., vol. 91, iss. 2, pp. 312-345, 1990. · Zbl 0711.46036
[35] D. Preiss and G. Speight, ”Differentiability of Lipschitz functions in Lebesgue null sets,” Invent. Math., vol. 199, iss. 2, pp. 517-559, 2015. · Zbl 1317.26011
[36] F. Rindler, ”Lower semicontinuity for integral functionals in the space of functions of bounded deformation via rigidity and Young measures,” Arch. Ration. Mech. Anal., vol. 202, iss. 1, pp. 63-113, 2011. · Zbl 1258.49014
[37] F. Rindler, ”Lower semicontinuity and Young measures in BV without Alberti’s rank-one theorem,” Adv. Calc. Var., vol. 5, iss. 2, pp. 127-159, 2012. · Zbl 1239.49018
[38] F. Rindler, ”Directional oscillations, concentrations, and compensated compactness via microlocal compactness forms,” Arch. Ration. Mech. Anal., vol. 215, iss. 1, pp. 1-63, 2015. · Zbl 1327.35004
[39] F. Rindler, ”A local proof for the characterization of Young measures generated by sequences in BV,” J. Funct. Anal., vol. 266, iss. 11, pp. 6335-6371, 2014. · Zbl 1305.49019
[40] A. Schioppa, ”Poincaré inequalities for mutually singular measures,” Anal. Geom. Metr. Spaces, vol. 3, pp. 40-45, 2015. · Zbl 1310.26017
[41] E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton, NJ: Princeton Univ. Press, 1993, vol. 43. · Zbl 0821.42001
[42] L. Tartar, ”Compensated compactness and applications to partial differential equations,” in Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, Vol. IV, Boston: Pitman, 1979, vol. 39, pp. 136-212. · Zbl 0437.35004
[43] L. Tartar, ”The compensated compactness method applied to systems of conservation laws,” in Systems of Nonlinear Partial Differential Equations, Reidel, Dordrecht, 1983, vol. 111, pp. 263-285. · Zbl 0536.35003
[44] R. Temam, Problèmes Mathématiques en Plasticité, Montrouge: Gauthier-Villars, 1983, vol. 12. · Zbl 0547.73026
[45] R. Temam and G. Strang, ”Functions of bounded deformation,” Arch. Rational Mech. Anal., vol. 75, iss. 1, pp. 7-21, 1980/81. · Zbl 0472.73031
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