American Institute of Mathematical Sciences

Advanced
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

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 & 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

show all references

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

[1]

Alexander Mielke. Weak-convergence methods for Hamiltonian multiscale problems. Discrete & Continuous Dynamical Systems - A, 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 & Continuous Dynamical Systems - S, 2012, 5 (4) : 779-788. doi: 10.3934/dcdss.2012.5.779
[3]

Yan Tang. Convergence analysis of a new iterative algorithm for solving split variational inclusion problems. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-20. doi: 10.3934/jimo.2018187
[4]

Alain Bensoussan, Miroslav Bulíček, Jens Frehse. Existence and compactness for weak solutions to Bellman systems with critical growth. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1729-1750. doi: 10.3934/dcdsb.2012.17.1729
[5]

Piotr Kowalski. The existence of a solution for Dirichlet boundary value problem for a Duffing type differential inclusion. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2569-2580. doi: 10.3934/dcdsb.2014.19.2569
[6]

T. Caraballo, J. A. Langa, J. Valero. Structure of the pullback attractor for a non-autonomous scalar differential inclusion. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 979-994. doi: 10.3934/dcdss.2016037
[7]

Ziqing Yuana, Jianshe Yu. Existence and multiplicity of nontrivial solutions of biharmonic equations via differential inclusion. Communications on Pure & Applied Analysis, 2020, 19 (1) : 391-405. doi: 10.3934/cpaa.2020020
[8]

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
[9]

Roberto Livrea, Salvatore A. Marano. A min-max principle for non-differentiable functions with a weak compactness condition. Communications on Pure & Applied Analysis, 2009, 8 (3) : 1019-1029. doi: 10.3934/cpaa.2009.8.1019
[10]

Stefan Kindermann, Antonio Leitão. Convergence rates for Kaczmarz-type regularization methods. Inverse Problems & Imaging, 2014, 8 (1) : 149-172. doi: 10.3934/ipi.2014.8.149
[11]

Clara Carlota, António Ornelas. The DuBois-Reymond differential inclusion for autonomous optimal control problems with pointwise-constrained derivatives. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 467-484. doi: 10.3934/dcds.2011.29.467
[12]

Antonia Chinnì, Roberto Livrea. Multiple solutions for a Neumann-type differential inclusion problem involving the $p(\cdot)$-Laplacian. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 753-764. doi: 10.3934/dcdss.2012.5.753
[13]

Dina Kalinichenko, Volker Reitmann, Sergey Skopinov. Asymptotic behavior of solutions to a coupled system of Maxwell's equations and a controlled differential inclusion. Conference Publications, 2013, 2013 (special) : 407-414. doi: 10.3934/proc.2013.2013.407
[14]

Jan Čermák, Jana Hrabalová. Delay-dependent stability criteria for neutral delay differential and difference equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4577-4588. doi: 10.3934/dcds.2014.34.4577
[15]

Suqi Ma, Zhaosheng Feng, Qishao Lu. A two-parameter geometrical criteria for delay differential equations. Discrete & Continuous Dynamical Systems - B, 2008, 9 (2) : 397-413. doi: 10.3934/dcdsb.2008.9.397
[16]

Masakatsu Suzuki, Hideaki Matsunaga. Stability criteria for a class of linear differential equations with off-diagonal delays. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1381-1391. doi: 10.3934/dcds.2009.24.1381
[17]

Xiaomeng Li, Qiang Xu, Ailing Zhu. Weak Galerkin mixed finite element methods for parabolic equations with memory. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 513-531. doi: 10.3934/dcdss.2019034
[18]

Hui Liang, Hermann Brunner. Collocation methods for differential equations with piecewise linear delays. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1839-1857. doi: 10.3934/cpaa.2012.11.1839
[19]

Joseph A. Connolly, Neville J. Ford. Comparison of numerical methods for fractional differential equations. Communications on Pure & Applied Analysis, 2006, 5 (2) : 289-307. doi: 10.3934/cpaa.2006.5.289
[20]

Qingguang Guan, Max Gunzburger. Stability and convergence of time-stepping methods for a nonlocal model for diffusion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1315-1335. doi: 10.3934/dcdsb.2015.20.1315

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences