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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), 129-147. |
[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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