Compactness versus regularity in the calculus of variations

Previous Article
Vector-valued obstacle problems for non-local energies
DCDS-B Home
This Issue
Next Article
Transport processes with coagulation and strong fragmentation

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
Related Papers

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.

[1]	Alexander Mielke. Weak-convergence methods for Hamiltonian multiscale problems. Discrete and Continuous Dynamical Systems, 2008, 20 (1) : 53-79. doi: 10.3934/dcds.2008.20.53
[2]	Francesca Faraci, Antonio Iannizzotto. Three nonzero periodic solutions for a differential inclusion. Discrete and Continuous Dynamical Systems - S, 2012, 5 (4) : 779-788. doi: 10.3934/dcdss.2012.5.779
[3]	Yongjian Liu, Zhenhai Liu, Dumitru Motreanu. Differential inclusion problems with convolution and discontinuous nonlinearities. Evolution Equations and Control Theory, 2020, 9 (4) : 1057-1071. doi: 10.3934/eect.2020056
[4]	Alain Bensoussan, Miroslav Bulíček, Jens Frehse. Existence and compactness for weak solutions to Bellman systems with critical growth. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 1729-1750. doi: 10.3934/dcdsb.2012.17.1729
[5]	Dang Van Hieu, Le Dung Muu, Pham Kim Quy. New iterative regularization methods for solving split variational inclusion problems. Journal of Industrial and Management Optimization, 2023, 19 (1) : 300-320. doi: 10.3934/jimo.2021185
[6]	Yan Tang. Convergence analysis of a new iterative algorithm for solving split variational inclusion problems. Journal of Industrial and Management Optimization, 2020, 16 (2) : 945-964. doi: 10.3934/jimo.2018187
[7]	Piotr Kowalski. The existence of a solution for Dirichlet boundary value problem for a Duffing type differential inclusion. Discrete and Continuous Dynamical Systems - B, 2014, 19 (8) : 2569-2580. doi: 10.3934/dcdsb.2014.19.2569
[8]	T. Caraballo, J. A. Langa, J. Valero. Structure of the pullback attractor for a non-autonomous scalar differential inclusion. Discrete and Continuous Dynamical Systems - S, 2016, 9 (4) : 979-994. doi: 10.3934/dcdss.2016037
[9]	Ziqing Yuana, Jianshe Yu. Existence and multiplicity of nontrivial solutions of biharmonic equations via differential inclusion. Communications on Pure and Applied Analysis, 2020, 19 (1) : 391-405. doi: 10.3934/cpaa.2020020
[10]	Jens Lorenz, Wilberclay G. Melo, Suelen C. P. de Souza. Regularity criteria for weak solutions of the Magneto-micropolar equations. Electronic Research Archive, 2021, 29 (1) : 1625-1639. doi: 10.3934/era.2020083
[11]	Tomoyuki Suzuki. Regularity criteria in weak spaces in terms of the pressure to the MHD equations. Conference Publications, 2011, 2011 (Special) : 1335-1343. doi: 10.3934/proc.2011.2011.1335
[12]	Shaoqiang Shang, Yunan Cui. Weak approximative compactness of hyperplane and Asplund property in Musielak-Orlicz-Bochner function spaces. Electronic Research Archive, 2020, 28 (1) : 327-346. doi: 10.3934/era.2020019
[13]	Roberto Livrea, Salvatore A. Marano. A min-max principle for non-differentiable functions with a weak compactness condition. Communications on Pure and Applied Analysis, 2009, 8 (3) : 1019-1029. doi: 10.3934/cpaa.2009.8.1019
[14]	Stefan Kindermann, Antonio Leitão. Convergence rates for Kaczmarz-type regularization methods. Inverse Problems and Imaging, 2014, 8 (1) : 149-172. doi: 10.3934/ipi.2014.8.149
[15]	Jan Čermák, Jana Hrabalová. Delay-dependent stability criteria for neutral delay differential and difference equations. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4577-4588. doi: 10.3934/dcds.2014.34.4577
[16]	Suqi Ma, Zhaosheng Feng, Qishao Lu. A two-parameter geometrical criteria for delay differential equations. Discrete and Continuous Dynamical Systems - B, 2008, 9 (2) : 397-413. doi: 10.3934/dcdsb.2008.9.397
[17]	Masakatsu Suzuki, Hideaki Matsunaga. Stability criteria for a class of linear differential equations with off-diagonal delays. Discrete and Continuous Dynamical Systems, 2009, 24 (4) : 1381-1391. doi: 10.3934/dcds.2009.24.1381
[18]	Zengyun Wang, Jinde Cao, Zuowei Cai, Lihong Huang. Finite-time stability of impulsive differential inclusion: Applications to discontinuous impulsive neural networks. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2677-2692. doi: 10.3934/dcdsb.2020200
[19]	Clara Carlota, António Ornelas. The DuBois-Reymond differential inclusion for autonomous optimal control problems with pointwise-constrained derivatives. Discrete and Continuous Dynamical Systems, 2011, 29 (2) : 467-484. doi: 10.3934/dcds.2011.29.467
[20]	Antonia Chinnì, Roberto Livrea. Multiple solutions for a Neumann-type differential inclusion problem involving the $p(\cdot)$-Laplacian. Discrete and Continuous Dynamical Systems - S, 2012, 5 (4) : 753-764. doi: 10.3934/dcdss.2012.5.753

2021 Impact Factor: 1.497

Tools

Metrics

Other articles
by authors

on AIMS
Daniel Faraco Jan Kristensen
on Google Scholar
Daniel Faraco Jan Kristensen

[Back to Top]