# American Institute of Mathematical Sciences

March  2012, 17(2): 473-485. doi: 10.3934/dcdsb.2012.17.473

## Compactness versus regularity in the calculus of variations

 1 Department of Mathematics, Universidad Autónoma de Madrid, and Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM, Campus de Cantoblanco, Madrid, 28049, Spain 2 Mathematical Institute, 24–29 St Giles’, University of Oxford, OX1 3LB Oxford, United Kingdom

Received  September 2010 Revised  February 2011 Published  December 2011

In this note we take the view that compactness in $L^p$ can be seen quantitatively on a scale of fractional Sobolev type spaces. To accommodate this viewpoint one must work on a scale of spaces, where the degree of differentiability is measured, not by a power function, but by an arbitrary function that decays to zero with its argument. In this context we provide new $L^p$ compactness criteria that were motivated by recent regularity results for minimizers of quasiconvex integrals. We also show how rigidity results for approximate solutions to certain differential inclusions follow from the Riesz--Kolmogorov compactness criteria.
Citation: Daniel Faraco, Jan Kristensen. Compactness versus regularity in the calculus of variations. Discrete & Continuous Dynamical Systems - B, 2012, 17 (2) : 473-485. doi: 10.3934/dcdsb.2012.17.473
##### References:
 [1] J. J. Alibert and G. Bouchitté, Non-uniform integrability and generalized Young measures,, J. Convex Anal., 4 (1997), 129.   Google Scholar [2] K. Astala and D. Faraco, Quasiregular mappings and Young measures,, Proc. Roy. Soc. Edinb. Sect. A, 132 (2002), 1045.   Google Scholar [3] J. M. Ball and R. D. James, Fine phase mixtures as minimizers of energy,, Arch. Ration. Mech. Anal., 100 (1987), 13.  doi: 10.1007/BF00281246.  Google Scholar [4] J. M. Ball and R. D. James, Proposed experimental tests of a theory of fine microstructure and the two-well problem,, Phil. Trans. R. Soc. Lond. A, 338 (1992), 389.  doi: 10.1098/rsta.1992.0013.  Google Scholar [5] M. Chlebik and B. Kirchheim, Rigidity for the four gradient problem,, J. Reine Angew. Math., 551 (2002), 1.   Google Scholar [6] J. R. Dorronsoro, A characterization of potential spaces,, Proc. Amer. Math. Soc., 95 (1985), 21.  doi: 10.1090/S0002-9939-1985-0796440-3.  Google Scholar [7] D.Faraco, Tartar conjecture and Beltrami operators,, Michigan Math. J., 52 (2004), 83.  doi: 10.1307/mmj/1080837736.  Google Scholar [8] D. Faraco and L. Székelyhidi, Tartar's conjecture and localization of the quasiconvex hull in $\mathbbR^{2\times 2}$,, Acta Math., 200 (2008), 279.  doi: 10.1007/s11511-008-0028-1.  Google Scholar [9] G. Friesecke, R. D. James and S. Müller, A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity,, Comm. Pure Appl. Math., 55 (2002), 1461.  doi: 10.1002/cpa.10048.  Google Scholar [10] M. Gromov, "Partial Differential Relations,", Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 9 (1986).   Google Scholar [11] P. Hajłasz, P. Koskela and H. Tuominen, Sobolev embeddings, extensions and measure density condition,, J. Funct. Anal., 254 (2008), 1217.  doi: 10.1016/j.jfa.2007.11.020.  Google Scholar [12] H. Hanche-Olsen and H. Holden, The Kolmogorov-Riesz compactness theorem,, Expositiones Mathematicae, (2010).  doi: 10.1016/j.exmath.2010.03.001.  Google Scholar [13] T. Iwaniec and G. Martin, "Geometric Function Theory and Non-Linear Analysis,", Oxford Mathematical Monographs, (2001).   Google Scholar [14] S. Janson, Generalizations of Lipschitz spaces and applications to Hardy spaces and bounded mean oscillation,, Duke Math. J., 47 (1980), 959.  doi: 10.1215/S0012-7094-80-04755-9.  Google Scholar [15] B. Kirchheim, "Rigidity and Geometry of Microstructures,", Habilitation Thesis, (2003).   Google Scholar [16] B. Kirchheim, S. Müller and V. Šverák, Studying nonlinear PDE by geometry in matrix space,, in, (2003), 347.   Google Scholar [17] J. Kristensen and G. Mingione, The singular set of Lipschitzian minima of multiple integrals,, Arch. Ration. Mech. Anal., 184 (2007), 341.  doi: 10.1007/s00205-006-0036-2.  Google Scholar [18] J. Kristensen and F. Rindler, Characterization of generalized gradient Young measures generated by sequences in $W^{1,1}$ and $BV$,, Arch. Ration. Mech. Anal., 197 (2010), 539.  doi: 10.1007/s00205-009-0287-9.  Google Scholar [19] S. Müller, A sharp version of Zhang's theorem on truncating sequences of gradients,, Trans. Amer. Math. Soc., 351 (1999), 4585.  doi: 10.1090/S0002-9947-99-02520-9.  Google Scholar [20] S. Müller, Variational models for microstructure and phase transitions,, in, 1713 (1999), 85.   Google Scholar [21] S. Müller and V. Šverák, Attainment results for the two-well problem by convex integration,, in, (1996), 239.   Google Scholar [22] S. Müller and V. Šverák, Convex integration for Lipschitz mappings and counterexamples to regularity,, Ann. Math. (2), 157 (2003), 715.   Google Scholar [23] V. Šverák, Rank-one convexity does not imply quasiconvexity,, Proc. Roy. Soc. Edinb. Sect. A, 120 (1992), 185.  doi: 10.1017/S0308210500015080.  Google Scholar [24] V. Šverák, On Tartar's conjecture,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 10 (1993), 405.   Google Scholar [25] V. Šverák, On the problem of two wells,, in, 54 (1993), 183.   Google Scholar [26] L. Székelyhidi, Jr., The regularity of critical points of polyconvex functionals,, Arch. Ration. Mech. Anal., 172 (2004), 133.  doi: 10.1007/s00205-003-0300-7.  Google Scholar [27] L. Székelyhidi, Jr., Rank-one convex hulls in $\mathbbR^{2\times 2}$,, Calc. Var. Partial Diff. Eq., 22 (2005), 253.   Google Scholar [28] L. Tartar, Compensated compactness and applications to partial differential equations,, in, 39 (1979), 136.   Google Scholar [29] K. Zhang, A construction of quasiconvex functions with linear growth at infinity,, Ann. Sc. Norm. Sup. Pisa Cl. Sci. (4), 19 (1992), 313.   Google Scholar

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##### References:
 [1] J. J. Alibert and G. Bouchitté, Non-uniform integrability and generalized Young measures,, J. Convex Anal., 4 (1997), 129.   Google Scholar [2] K. Astala and D. Faraco, Quasiregular mappings and Young measures,, Proc. Roy. Soc. Edinb. Sect. A, 132 (2002), 1045.   Google Scholar [3] J. M. Ball and R. D. James, Fine phase mixtures as minimizers of energy,, Arch. Ration. Mech. Anal., 100 (1987), 13.  doi: 10.1007/BF00281246.  Google Scholar [4] J. M. Ball and R. D. James, Proposed experimental tests of a theory of fine microstructure and the two-well problem,, Phil. Trans. R. Soc. Lond. A, 338 (1992), 389.  doi: 10.1098/rsta.1992.0013.  Google Scholar [5] M. Chlebik and B. Kirchheim, Rigidity for the four gradient problem,, J. Reine Angew. Math., 551 (2002), 1.   Google Scholar [6] J. R. Dorronsoro, A characterization of potential spaces,, Proc. Amer. Math. Soc., 95 (1985), 21.  doi: 10.1090/S0002-9939-1985-0796440-3.  Google Scholar [7] D.Faraco, Tartar conjecture and Beltrami operators,, Michigan Math. J., 52 (2004), 83.  doi: 10.1307/mmj/1080837736.  Google Scholar [8] D. Faraco and L. Székelyhidi, Tartar's conjecture and localization of the quasiconvex hull in $\mathbbR^{2\times 2}$,, Acta Math., 200 (2008), 279.  doi: 10.1007/s11511-008-0028-1.  Google Scholar [9] G. Friesecke, R. D. James and S. Müller, A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity,, Comm. Pure Appl. Math., 55 (2002), 1461.  doi: 10.1002/cpa.10048.  Google Scholar [10] M. Gromov, "Partial Differential Relations,", Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 9 (1986).   Google Scholar [11] P. Hajłasz, P. Koskela and H. Tuominen, Sobolev embeddings, extensions and measure density condition,, J. Funct. Anal., 254 (2008), 1217.  doi: 10.1016/j.jfa.2007.11.020.  Google Scholar [12] H. Hanche-Olsen and H. Holden, The Kolmogorov-Riesz compactness theorem,, Expositiones Mathematicae, (2010).  doi: 10.1016/j.exmath.2010.03.001.  Google Scholar [13] T. Iwaniec and G. Martin, "Geometric Function Theory and Non-Linear Analysis,", Oxford Mathematical Monographs, (2001).   Google Scholar [14] S. Janson, Generalizations of Lipschitz spaces and applications to Hardy spaces and bounded mean oscillation,, Duke Math. J., 47 (1980), 959.  doi: 10.1215/S0012-7094-80-04755-9.  Google Scholar [15] B. Kirchheim, "Rigidity and Geometry of Microstructures,", Habilitation Thesis, (2003).   Google Scholar [16] B. Kirchheim, S. Müller and V. Šverák, Studying nonlinear PDE by geometry in matrix space,, in, (2003), 347.   Google Scholar [17] J. Kristensen and G. Mingione, The singular set of Lipschitzian minima of multiple integrals,, Arch. Ration. Mech. Anal., 184 (2007), 341.  doi: 10.1007/s00205-006-0036-2.  Google Scholar [18] J. Kristensen and F. Rindler, Characterization of generalized gradient Young measures generated by sequences in $W^{1,1}$ and $BV$,, Arch. Ration. Mech. Anal., 197 (2010), 539.  doi: 10.1007/s00205-009-0287-9.  Google Scholar [19] S. Müller, A sharp version of Zhang's theorem on truncating sequences of gradients,, Trans. Amer. Math. Soc., 351 (1999), 4585.  doi: 10.1090/S0002-9947-99-02520-9.  Google Scholar [20] S. Müller, Variational models for microstructure and phase transitions,, in, 1713 (1999), 85.   Google Scholar [21] S. Müller and V. Šverák, Attainment results for the two-well problem by convex integration,, in, (1996), 239.   Google Scholar [22] S. Müller and V. Šverák, Convex integration for Lipschitz mappings and counterexamples to regularity,, Ann. Math. (2), 157 (2003), 715.   Google Scholar [23] V. Šverák, Rank-one convexity does not imply quasiconvexity,, Proc. Roy. Soc. Edinb. Sect. A, 120 (1992), 185.  doi: 10.1017/S0308210500015080.  Google Scholar [24] V. Šverák, On Tartar's conjecture,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 10 (1993), 405.   Google Scholar [25] V. Šverák, On the problem of two wells,, in, 54 (1993), 183.   Google Scholar [26] L. Székelyhidi, Jr., The regularity of critical points of polyconvex functionals,, Arch. Ration. Mech. Anal., 172 (2004), 133.  doi: 10.1007/s00205-003-0300-7.  Google Scholar [27] L. Székelyhidi, Jr., Rank-one convex hulls in $\mathbbR^{2\times 2}$,, Calc. Var. Partial Diff. Eq., 22 (2005), 253.   Google Scholar [28] L. Tartar, Compensated compactness and applications to partial differential equations,, in, 39 (1979), 136.   Google Scholar [29] K. Zhang, A construction of quasiconvex functions with linear growth at infinity,, Ann. Sc. Norm. Sup. Pisa Cl. Sci. (4), 19 (1992), 313.   Google Scholar
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