-
Previous Article
Vector-valued obstacle problems for non-local energies
- DCDS-B Home
- This Issue
-
Next Article
Transport processes with coagulation and strong fragmentation
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 |
References:
[1] |
J. J. Alibert and G. Bouchitté, Non-uniform integrability and generalized Young measures,, J. Convex Anal., 4 (1997), 129.
|
[2] |
K. Astala and D. Faraco, Quasiregular mappings and Young measures,, Proc. Roy. Soc. Edinb. Sect. A, 132 (2002), 1045.
|
[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. |
[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. |
[5] |
M. Chlebik and B. Kirchheim, Rigidity for the four gradient problem,, J. Reine Angew. Math., 551 (2002), 1.
|
[6] |
J. R. Dorronsoro, A characterization of potential spaces,, Proc. Amer. Math. Soc., 95 (1985), 21.
doi: 10.1090/S0002-9939-1985-0796440-3. |
[7] |
D.Faraco, Tartar conjecture and Beltrami operators,, Michigan Math. J., 52 (2004), 83.
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.
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.
doi: 10.1002/cpa.10048. |
[10] |
M. Gromov, "Partial Differential Relations,", Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 9 (1986).
|
[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. |
[12] |
H. Hanche-Olsen and H. Holden, The Kolmogorov-Riesz compactness theorem,, Expositiones Mathematicae, (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, (2001).
|
[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. |
[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.
|
[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. |
[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. |
[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. |
[20] |
S. Müller, Variational models for microstructure and phase transitions,, in, 1713 (1999), 85.
|
[21] |
S. Müller and V. Šverák, Attainment results for the two-well problem by convex integration,, in, (1996), 239.
|
[22] |
S. Müller and V. Šverák, Convex integration for Lipschitz mappings and counterexamples to regularity,, Ann. Math. (2), 157 (2003), 715.
|
[23] |
V. Šverák, Rank-one convexity does not imply quasiconvexity,, Proc. Roy. Soc. Edinb. Sect. A, 120 (1992), 185.
doi: 10.1017/S0308210500015080. |
[24] |
V. Šverák, On Tartar's conjecture,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 10 (1993), 405.
|
[25] |
V. Šverák, On the problem of two wells,, in, 54 (1993), 183.
|
[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. |
[27] |
L. Székelyhidi, Jr., Rank-one convex hulls in $\mathbbR^{2\times 2}$,, Calc. Var. Partial Diff. Eq., 22 (2005), 253.
|
[28] |
L. Tartar, Compensated compactness and applications to partial differential equations,, in, 39 (1979), 136.
|
[29] |
K. Zhang, A construction of quasiconvex functions with linear growth at infinity,, Ann. Sc. Norm. Sup. Pisa Cl. Sci. (4), 19 (1992), 313.
|
show all references
References:
[1] |
J. J. Alibert and G. Bouchitté, Non-uniform integrability and generalized Young measures,, J. Convex Anal., 4 (1997), 129.
|
[2] |
K. Astala and D. Faraco, Quasiregular mappings and Young measures,, Proc. Roy. Soc. Edinb. Sect. A, 132 (2002), 1045.
|
[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. |
[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. |
[5] |
M. Chlebik and B. Kirchheim, Rigidity for the four gradient problem,, J. Reine Angew. Math., 551 (2002), 1.
|
[6] |
J. R. Dorronsoro, A characterization of potential spaces,, Proc. Amer. Math. Soc., 95 (1985), 21.
doi: 10.1090/S0002-9939-1985-0796440-3. |
[7] |
D.Faraco, Tartar conjecture and Beltrami operators,, Michigan Math. J., 52 (2004), 83.
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.
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.
doi: 10.1002/cpa.10048. |
[10] |
M. Gromov, "Partial Differential Relations,", Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 9 (1986).
|
[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. |
[12] |
H. Hanche-Olsen and H. Holden, The Kolmogorov-Riesz compactness theorem,, Expositiones Mathematicae, (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, (2001).
|
[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. |
[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.
|
[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. |
[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. |
[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. |
[20] |
S. Müller, Variational models for microstructure and phase transitions,, in, 1713 (1999), 85.
|
[21] |
S. Müller and V. Šverák, Attainment results for the two-well problem by convex integration,, in, (1996), 239.
|
[22] |
S. Müller and V. Šverák, Convex integration for Lipschitz mappings and counterexamples to regularity,, Ann. Math. (2), 157 (2003), 715.
|
[23] |
V. Šverák, Rank-one convexity does not imply quasiconvexity,, Proc. Roy. Soc. Edinb. Sect. A, 120 (1992), 185.
doi: 10.1017/S0308210500015080. |
[24] |
V. Šverák, On Tartar's conjecture,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 10 (1993), 405.
|
[25] |
V. Šverák, On the problem of two wells,, in, 54 (1993), 183.
|
[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. |
[27] |
L. Székelyhidi, Jr., Rank-one convex hulls in $\mathbbR^{2\times 2}$,, Calc. Var. Partial Diff. Eq., 22 (2005), 253.
|
[28] |
L. Tartar, Compensated compactness and applications to partial differential equations,, in, 39 (1979), 136.
|
[29] |
K. Zhang, A construction of quasiconvex functions with linear growth at infinity,, Ann. Sc. Norm. Sup. Pisa Cl. Sci. (4), 19 (1992), 313.
|
[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
Other articles
by authors
[Back to Top]