Directional Oscillations, Concentrations, and Compensated Compactness via Microlocal Compactness Forms

Alibert J.J., Bouchitté G.: Non-uniform integrability and generalized Young measures. J. Convex Anal. 4, 129–147 (1997) Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free-Discontinuity Problems. Oxford Mathematical Monographs. Oxford University Press, 2000 Antonić, N., Mitrović, D.: H-distributions: an extension of H-measures to an L p-L q setting. Abstr. Appl. Anal. Article ID 901,084 (2011) Ball J.M., James R.D.: Fine phase mixtures as minimizers of energy. Arch. Ration. Mech. Anal. 100, 13–52 (1987) Coifman R.R., Rochberg R., Weiss G.: Factorization theorems for Hardy spaces in several variables. Ann. Math. 103, 611–635 (1976) Dacorogna, B.: Direct Methods in the Calculus of Variations. Applied Mathematical Sciences, Vol. 78, 2nd edn. Springer, Berlin, 2008 De Lellis, C., Székelyhidi Jr., L.: The Euler equations as a differential inclusion. Ann. Math. 170, 1417–1436 (2009) Diestel, J., Uhl, Jr., J.J.: Vector measures, Mathematical Surveys, Vol. 15. American Mathematical Society, Providence, 1977 DiPerna R.J., Majda A.J.: Oscillations and concentrations in weak solutions of the incompressible fluid equations. Commun. Math. Phys. 108, 667–689 (1987) Dunford N.J., Schwartz J.T.: Linear Operators I: General Theory. Interscience Publishers, New York (1958) Fermanian Kammerer, C., Gérard, P.: A Landau–Zener formula for non-degenerated involutive codimension 3 crossings. Ann. Henri Poincaré 4, 513–552 (2003) Fonseca, I., Leoni, G.: Modern Methods in the Calculus of Variations: L p Spaces. Springer, Berlin, 2007 Fonseca, I., Müller, S.: \({{\mathcal{A}}}\)-quasiconvexity, lower semicontinuity, and Young measures. SIAM J. Math. Anal. 30(6), 1355–1390 (1999) Gérard, P.: Compacité par compensation et régularité deux-microlocale. In: Séminaire Équations aux Dérivées Partielles. École Polytechnique, Palaiseau, 1988–1989 Gérard P.: Microlocal defect measures. Commun. Partial Differ. Equ. 16, 1761–1794 (1991) Grafakos, L.: Classical Fourier Analysis. Graduate Texts in Mathematics, Vol. 249, 2nd edn. Springer, Berlin, 2008 Hörmander, L.: The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis. Grundlehren der mathematischen Wissenschaften, Vol. 256, 2nd edn. Springer, Berlin, 1990 Hörmander, L.: The Analysis of Linear Partial Differential Operators. III. Grundlehren der mathematischen Wissenschaften, Vol. 274. Springer, Berlin, 2007 Joly, J.L., Métivier, G., Rauch, J.: Trilinear compensated compactness and nonlinear geometric optics. Ann. Math. 142, 121–169 (1995) 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.: Rigidity and Geometry of Microstructures. Lecture Notes 16. Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, 2003 Kristensen, J., Rindler, F.: Characterization of generalized gradient Young measures generated by sequences in W1,1 and BV. Arch. Ration. Mech. Anal. 197, 539–598 (2010) [Erratum: 203, 693–700 (2012)] Kružík, M., Roubícek T.: On the measures of DiPerna and Majda. Math. Bohem. 122, 383–399 (1997) McLaughlin, D., Papanicolaou, G., Tartar, L.: Weak limits of semilinear hyperbolic systems with oscillating data. Macroscopic Modelling of Turbulent Flows (Nice, 1984). Lecture Notes in Physics, Vol. 230. Springer, Berlin, 277–289, 1985 Mielke A.: Flow properties for Young-measure solutions of semilinear hyperbolic problems. Proc. Roy. Soc. Edinb. Sect. A 129, 85–123 (1999) Müller S.: Rank-one convexity implies quasiconvexity on diagonal matrices. Int. Math. Res. Not. 20, 1087–1095 (1999) Müller, S.: Variational models for microstructure and phase transitions. Calculus of Variations and Geometric Evolution Problems (Cetraro, 1996). Lecture Notes in Mathematics, Vol. 1713, pp. 85–210. Springer (1999) Murat F.: Compacité par compensation. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 5, 489–507 (1978) Murat, F.: Compacité par compensation. II. Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis (Rome, 1978). Pitagora, 245–256, 1979 Murat, F.: Compacité par compensation: condition nécessaire et suffisante de continuité faible sous une hypothèse de rang constant. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 8, 69–102 (1981) 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) Ryan, R.A.: Introduction to Tensor Products of Banach Spaces. Springer Monographs in Mathematics. Springer, Berlin, 2002 Stefanov, A.: Pseudodifferential operators with rough symbols. J. Fourier Anal. Appl. 16, 97–128 (2010) Stein E.M.: Harmonic Analysis. Princeton University Press, Princeton (1993) Sychev, M.A.: Characterization of homogeneous gradient Young measures in case of arbitrary integrands. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 29, 531–548 (2000) Székelyhidi, L., Wiedemann, E.: Young measures generated by ideal incompressible fluid flows. Arch. Ration. Mech. Anal. 206, 333–366 (2012) Tartar, L.: Compensated compactness and applications to partial differential equations. Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, Vol. IV. Research Notes in Mathematics, Vol. 39. Pitman, 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 Tartar, L.: H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations. Proc. Roy. Soc. Edinb. Sect. A 115, 193–230 (1990) Tartar L.: Beyond Young measures. Meccanica 30, 505–526 (1995) Tartar, L.: An Introduction to Navier–Stokes Equation and Oceanography. Lecture Notes of the Unione Matematica Italiana, Vol. 1. Springer, Berlin, 2006 Tartar, L.: The General Theory of Homogenization. A Personalized Introduction. Lecture Notes of the Unione Matematica Italiana, Vol. 7. Springer, Berlin, 2009 Taylor, M.E.: Pseudodifferential Operators and Nonlinear PDE. Progress in Mathematics, Vol. 100. Birkhäuser, Basel, 1991 Taylor, M.E.: Partial Differential Equations I. Basic Theory, Applied Mathematical Sciences, Vol. 115, 2nd edn. Springer, Berlin, 2011 Taylor, M.E.: Partial Differential Equations II. Qualitative Studies of Linear Equations. Applied Mathematical Sciences, Vol. 116, 2nd edn. Springer, Berlin, 2011 Taylor, M.E.: Partial Differential Equations III. Nonlinear Equations. Applied Mathematical Sciences, Vol. 117, 2nd edn. Springer, Berlin, 2011 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. Math. 43, 84–103 (1942) Young L.C.: Generalized surfaces in the calculus of variations. II. Ann. Math. 43, 530–544 (1942) Young, L.C.: Lectures on the Calculus of Variations and Optimal Control Theory, 2nd edn. Chelsea, New York, 1980 (Reprinted by AMS Chelsea Publishing 2000)