Compactness versus regularity in the calculus of variations

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

Compactness versus regularity in the calculus of variations

Daniel Faraco ^1, and Jan Kristensen ^2,

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

Abstract
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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.

Keywords: Weak convergence methods, differential inclusion., compactness criteria.

Mathematics Subject Classification: Primary: 49J45; Secondary: 46E35, 46E3.

Citation: Daniel Faraco, Jan Kristensen. Compactness versus regularity in the calculus of variations. Discrete and 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), 129-147.
[2]	K. Astala and D. Faraco, Quasiregular mappings and Young measures, Proc. Roy. Soc. Edinb. Sect. A, 132 (2002), 1045-1056.
[3]	J. M. Ball and R. D. James, Fine phase mixtures as minimizers of energy, Arch. Ration. Mech. Anal., 100 (1987), 13-52. doi: 10.1007/BF00281246.
[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), 389-450. doi: 10.1098/rsta.1992.0013.
[5]	M. Chlebik and B. Kirchheim, Rigidity for the four gradient problem, J. Reine Angew. Math., 551 (2002), 1-9.
[6]	J. R. Dorronsoro, A characterization of potential spaces, Proc. Amer. Math. Soc., 95 (1985), 21-31. doi: 10.1090/S0002-9939-1985-0796440-3.
[7]	D.Faraco, Tartar conjecture and Beltrami operators, Michigan Math. J., 52 (2004), 83-104. doi: 10.1307/mmj/1080837736.
[8]	D. Faraco and L. Székelyhidi, Tartar's conjecture and localization of the quasiconvex hull in $\mathbbR^{2\times 2}$, Acta Math., 200 (2008), 279-305. doi: 10.1007/s11511-008-0028-1.
[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), 1461-1506. doi: 10.1002/cpa.10048.
[10]	M. Gromov, "Partial Differential Relations," Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 9, Springer-Verlag, Berlin, 1986.
[11]	P. Hajłasz, P. Koskela and H. Tuominen, Sobolev embeddings, extensions and measure density condition, J. Funct. Anal., 254 (2008), 1217-1234. doi: 10.1016/j.jfa.2007.11.020.
[12]	H. Hanche-Olsen and H. Holden, The Kolmogorov-Riesz compactness theorem, Expositiones Mathematicae, in press, 2010. doi: 10.1016/j.exmath.2010.03.001.
[13]	T. Iwaniec and G. Martin, "Geometric Function Theory and Non-Linear Analysis," Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2001.
[14]	S. Janson, Generalizations of Lipschitz spaces and applications to Hardy spaces and bounded mean oscillation, Duke Math. J., 47 (1980), 959-982. doi: 10.1215/S0012-7094-80-04755-9.
[15]	B. Kirchheim, "Rigidity and Geometry of Microstructures," Habilitation Thesis, University of Leipzig, 2003.
[16]	B. Kirchheim, S. Müller and V. Šverák, Studying nonlinear PDE by geometry in matrix space, in "Geometric Analysis and Nonlinear Partial Differential Equations" (eds. S. Hildebrandt and H. Karcher), Springer-Verlag, Berlin, (2003), 347-395.
[17]	J. Kristensen and G. Mingione, The singular set of Lipschitzian minima of multiple integrals, Arch. Ration. Mech. Anal., 184 (2007), 341-369. doi: 10.1007/s00205-006-0036-2.
[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), 539-598. doi: 10.1007/s00205-009-0287-9.
[19]	S. Müller, A sharp version of Zhang's theorem on truncating sequences of gradients, Trans. Amer. Math. Soc., 351 (1999), 4585-4597. doi: 10.1090/S0002-9947-99-02520-9.
[20]	S. Müller, Variational models for microstructure and phase transitions, in "Calculus of Variations and Geometric Evolution Problems" (Cetraro, 1996), Lecture Notes in Math., 1713, Springer, Berlin, (1999), 85-210.
[21]	S. Müller and V. Šverák, Attainment results for the two-well problem by convex integration, in "Geometric Analysis and the Calculus of Variations," 239-251, Internat. Press, Cambridge, MA, 1996.
[22]	S. Müller and V. Šverák, Convex integration for Lipschitz mappings and counterexamples to regularity, Ann. Math. (2), 157 (2003), 715-742.
[23]	V. Šverák, Rank-one convexity does not imply quasiconvexity, Proc. Roy. Soc. Edinb. Sect. A, 120 (1992), 185-189. doi: 10.1017/S0308210500015080.
[24]	V. Šverák, On Tartar's conjecture, Ann. Inst. H. Poincaré Anal. Non Linéaire, 10 (1993), 405-412.
[25]	V. Šverák, On the problem of two wells, in "Microstructures and Phase Transitions," IMA Vol. Math. Appl., 54, Springer, New York, (1993), 183-189.
[26]	L. Székelyhidi, Jr., The regularity of critical points of polyconvex functionals, Arch. Ration. Mech. Anal., 172 (2004), 133-152. doi: 10.1007/s00205-003-0300-7.
[27]	L. Székelyhidi, Jr., Rank-one convex hulls in $\mathbbR^{2\times 2}$, Calc. Var. Partial Diff. Eq., 22 (2005), 253-281.
[28]	L. Tartar, Compensated compactness and applications to partial differential equations, in "Nonlinear Analysis and Mechanics: Heriot-Watt Symposium," Vol. IV, Res. Notes in Math., 39, Pitman, Boston, Mass.-London, (1979), 136-212.
[29]	K. Zhang, A construction of quasiconvex functions with linear growth at infinity, Ann. Sc. Norm. Sup. Pisa Cl. Sci. (4), 19 (1992), 313-326.

show all references

References:

[1]	J. J. Alibert and G. Bouchitté, Non-uniform integrability and generalized Young measures, J. Convex Anal., 4 (1997), 129-147.
[2]	K. Astala and D. Faraco, Quasiregular mappings and Young measures, Proc. Roy. Soc. Edinb. Sect. A, 132 (2002), 1045-1056.
[3]	J. M. Ball and R. D. James, Fine phase mixtures as minimizers of energy, Arch. Ration. Mech. Anal., 100 (1987), 13-52. doi: 10.1007/BF00281246.
[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), 389-450. doi: 10.1098/rsta.1992.0013.
[5]	M. Chlebik and B. Kirchheim, Rigidity for the four gradient problem, J. Reine Angew. Math., 551 (2002), 1-9.
[6]	J. R. Dorronsoro, A characterization of potential spaces, Proc. Amer. Math. Soc., 95 (1985), 21-31. doi: 10.1090/S0002-9939-1985-0796440-3.
[7]	D.Faraco, Tartar conjecture and Beltrami operators, Michigan Math. J., 52 (2004), 83-104. doi: 10.1307/mmj/1080837736.
[8]	D. Faraco and L. Székelyhidi, Tartar's conjecture and localization of the quasiconvex hull in $\mathbbR^{2\times 2}$, Acta Math., 200 (2008), 279-305. doi: 10.1007/s11511-008-0028-1.
[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), 1461-1506. doi: 10.1002/cpa.10048.
[10]	M. Gromov, "Partial Differential Relations," Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 9, Springer-Verlag, Berlin, 1986.
[11]	P. Hajłasz, P. Koskela and H. Tuominen, Sobolev embeddings, extensions and measure density condition, J. Funct. Anal., 254 (2008), 1217-1234. doi: 10.1016/j.jfa.2007.11.020.
[12]	H. Hanche-Olsen and H. Holden, The Kolmogorov-Riesz compactness theorem, Expositiones Mathematicae, in press, 2010. doi: 10.1016/j.exmath.2010.03.001.
[13]	T. Iwaniec and G. Martin, "Geometric Function Theory and Non-Linear Analysis," Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2001.
[14]	S. Janson, Generalizations of Lipschitz spaces and applications to Hardy spaces and bounded mean oscillation, Duke Math. J., 47 (1980), 959-982. doi: 10.1215/S0012-7094-80-04755-9.
[15]	B. Kirchheim, "Rigidity and Geometry of Microstructures," Habilitation Thesis, University of Leipzig, 2003.
[16]	B. Kirchheim, S. Müller and V. Šverák, Studying nonlinear PDE by geometry in matrix space, in "Geometric Analysis and Nonlinear Partial Differential Equations" (eds. S. Hildebrandt and H. Karcher), Springer-Verlag, Berlin, (2003), 347-395.
[17]	J. Kristensen and G. Mingione, The singular set of Lipschitzian minima of multiple integrals, Arch. Ration. Mech. Anal., 184 (2007), 341-369. doi: 10.1007/s00205-006-0036-2.
[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), 539-598. doi: 10.1007/s00205-009-0287-9.
[19]	S. Müller, A sharp version of Zhang's theorem on truncating sequences of gradients, Trans. Amer. Math. Soc., 351 (1999), 4585-4597. doi: 10.1090/S0002-9947-99-02520-9.
[20]	S. Müller, Variational models for microstructure and phase transitions, in "Calculus of Variations and Geometric Evolution Problems" (Cetraro, 1996), Lecture Notes in Math., 1713, Springer, Berlin, (1999), 85-210.
[21]	S. Müller and V. Šverák, Attainment results for the two-well problem by convex integration, in "Geometric Analysis and the Calculus of Variations," 239-251, Internat. Press, Cambridge, MA, 1996.
[22]	S. Müller and V. Šverák, Convex integration for Lipschitz mappings and counterexamples to regularity, Ann. Math. (2), 157 (2003), 715-742.
[23]	V. Šverák, Rank-one convexity does not imply quasiconvexity, Proc. Roy. Soc. Edinb. Sect. A, 120 (1992), 185-189. doi: 10.1017/S0308210500015080.
[24]	V. Šverák, On Tartar's conjecture, Ann. Inst. H. Poincaré Anal. Non Linéaire, 10 (1993), 405-412.
[25]	V. Šverák, On the problem of two wells, in "Microstructures and Phase Transitions," IMA Vol. Math. Appl., 54, Springer, New York, (1993), 183-189.
[26]	L. Székelyhidi, Jr., The regularity of critical points of polyconvex functionals, Arch. Ration. Mech. Anal., 172 (2004), 133-152. doi: 10.1007/s00205-003-0300-7.
[27]	L. Székelyhidi, Jr., Rank-one convex hulls in $\mathbbR^{2\times 2}$, Calc. Var. Partial Diff. Eq., 22 (2005), 253-281.
[28]	L. Tartar, Compensated compactness and applications to partial differential equations, in "Nonlinear Analysis and Mechanics: Heriot-Watt Symposium," Vol. IV, Res. Notes in Math., 39, Pitman, Boston, Mass.-London, (1979), 136-212.
[29]	K. Zhang, A construction of quasiconvex functions with linear growth at infinity, Ann. Sc. Norm. Sup. Pisa Cl. Sci. (4), 19 (1992), 313-326.

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