Comparing two methods of resolving homogeneous differential inclusions

Calculus of Variations and Partial Differential Equations

Volumn 13, Issue 2, 2001, Pages 213-229

  Sychev, M A ^a,b  

^a MAX PLANCK INSTITUTE FOR MATHEMATICS IN THE SCIENCES   (Germany)

^b SOBOLEV INSTITUTE OF MATHEMATICS   (Russian Federation)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0345856295     PISSN: 09442669     EISSN: None     Source Type: Journal    
DOI: 10.1007/PL00009929     Document Type: Article

References (28)

1. Ball J.M., Currie J.C., Olver P.J.: Null Lagrangians, weak continuity, and variational problems of arbitrary order. J.Funct. Anal. 41 (1981), pp. 135-174.
2. Ball J.M., James R.D.: Fine mixtures as minimizers of energy. Arch. Rational Mech. Anal. 100 (1987), pp. 13-52.
3. Ball J.M., James R.D.: Proposed experimental tests of a theory of fine microstructure and the two-well problem. Phil. Trans. Roy.Soc. London A 338 (1992), pp.389-450.
4. Bhattacharya K., Firoozye N., James R.D., Kohn R.V.: Restrictions on microstructure. Proc. Roy. Soc. Edinburgh, 124 A (1994), pp.843-878.
5. Cellina A. On minima of functionals of gradient: necessary and sufficient conditions. Nonlinear Analysis TMA, 20 (1993), pp.337-347.
6. Dacorogna B., Marcellini P.: General existence theorems for Hamilton-Jacobi equations in the scalar and vectorial cases. Acta Mathematica 178 (1997), pp.1-37.
7. Dacorogna B., Marcellini P.: Cauchy-Dirichlet problem for first order nonlinear systems. J. Funct. Analysis 152 (1998), pp.404-446.
8. Dacorogna B., Marcellini P.: On the solvability of implicit nonlinear systems in the vectorial case. AMS series of Contemporary Mathematics, 1999 (to appear).
9. Dacorogna B., Marcellini P.: Implicit Partial Differential Equations. Birkhäuser, 1999.
10. Dacorogna B., Marcellini P.: Existence of minimizers for non quasiconvex integrals. Arch. Rational Mech. Anal., 131 (1995), pp.359-399.
11. Evans L.C., Gariepy R.F.: Measure theory and fine properties of functions. CRC Press, Boca Raton 1992.
12. Ekeland I., Temam R.: Convex analysis and variational problems. Amsterdam, North-Holland (1976).
13. Friesecke G. A necessary and sufficient condition for nonattainment and formation of microstructure almost everywhere in scalar variational problems. Proc.Roy.Soc.Edinburgh. Sect.A 124 (1994), pp.437-471.
14. Gromov M.: Partial differential relations. Springer, Berlin, 1986.
15. Guseinov F.V.: To the question of extensions of multi-dimensional variational problems. Izvestiya Mathematics, v. 50 (1986), N 1, pp.3-21.
16. 1-isometric imbeddings I, II, Indag. Math. 17 (1955), pp.545-556. pp. 683-689.
17. Matov V.I.: Investigation of the problem of multidimensional calculus of variations. Vestnik Moskovskogo Universiteta, Matematika 33 (1978), N 1, pp.61-69.
18. J. Matoušek, P. Plecháč. On functional separately convex hulls. J.Discrete Comput. Geom. 19 (1998), pp.105-130.
19. Morrey C.B.: Quasi-convexity and lower semicontinuity of multiple integrals. Pasific J. Math. 2 (1952), pp.25-53.
20. Müller S., Sychev M.: Optimal existence results for nonhomogeneous differential inclusions. Preprint 71, Max Planck Institute for Mathematics in Sciences, Leipzig, December 1999.
21. Müller S., Šverák V.: Attainment results for the two-well problem by convex integration. Edited by J. Jost, International Press, 1996, pp.239-251.
22. Müller S., Šverák V.: Unexpected solutions of first and second order partial differential equations. Documenta Math., Extra volume ICM 1998, II, pp.691-702.
23. Müller S.: A sharp version of Zhang's theorem on truncating sequences of gradients. TAMS 351 (1999), pp. 4585-4597.
24. 1-isometric imbeddings. Ann. Math. 63 (1954), pp.384-396.
25. Sychev M.: Characterization of homogeneous scalar variational problems solvable for all boundary data. Proc. Royal Soc. Edinburgh. Sect.A 130A (2000), pp.611-631.
26. Sychev M.: Characterization of homogeneous gradient Young measures in the case of arbitrary integrands. Annali della Scuola Normale Superiore, Classe di Scienze (to appear).
27. Sychev M.: Existence and relaxation results in special classes of deformations. Preprint No 17, Max-Planck-Institute for Mathematics in Sciences, Leipzig, February 2000.
28. Saks S.: Theory of the integral. Hafner, New York, 1937.