Topological degree, Jacobian determinants and relaxation

Irene Fonseca; Nicola Fusco; Paolo Marcellini

Bollettino dell'Unione Matematica Italiana (2005)

  • Volume: 8-B, Issue: 1, page 187-250
  • ISSN: 0392-4041

Abstract

top
A characterization of the total variation T ⁢ V ⁢ u , Ω of the Jacobian determinant det ⁡ D ⁢ u is obtained for some classes of functions u : Ω → R n outside the traditional regularity space W 1 , n ⁢ Ω ; R n . In particular, explicit formulas are deduced for functions that are locally Lipschitz continuous away from a given one point singularity x 0 ∈ Ω . Relations between T ⁢ V ⁢ u , Ω and the distributional determinant Det ⁢ D ⁢ u are established, and an integral representation is obtained for the relaxed energy of certain polyconvex functionals at maps u ∈ W 1 , p ⁢ Ω ; R n ∩ W 1 , ∞ ⁢ Ω x 0 ; R n .

How to cite

top

Fonseca, Irene, Fusco, Nicola, and Marcellini, Paolo. "Topological degree, Jacobian determinants and relaxation." Bollettino dell'Unione Matematica Italiana 8-B.1 (2005): 187-250. <http://eudml.org/doc/194572>.

@article{Fonseca2005,
abstract = {A characterization of the total variation $TV(u, \Omega)$ of the Jacobian determinant $\det Du$ is obtained for some classes of functions $u : \Omega \rightarrow \mathbb\{R\}^\{n\}$ outside the traditional regularity space $W^\{1, n\}(\Omega; \mathbb\{R\}^\{n\})$. In particular, explicit formulas are deduced for functions that are locally Lipschitz continuous away from a given one point singularity $x_\{0\}\in \Omega$. Relations between $TV(u, \Omega)$ and the distributional determinant $\text\{Det\}\, Du$ are established, and an integral representation is obtained for the relaxed energy of certain polyconvex functionals at maps $u\in W^\{1, p\}(\Omega; \mathbb\{R\}^\{n\})\cap W^\{1, \infty\}(\Omega\backslash \\{x_0\\}; \mathbb\{R\}^\{n\})$.},
author = {Fonseca, Irene, Fusco, Nicola, Marcellini, Paolo},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {2},
number = {1},
pages = {187-250},
publisher = {Unione Matematica Italiana},
title = {Topological degree, Jacobian determinants and relaxation},
url = {http://eudml.org/doc/194572},
volume = {8-B},
year = {2005},
}

TY - JOUR
AU - Fonseca, Irene
AU - Fusco, Nicola
AU - Marcellini, Paolo
TI - Topological degree, Jacobian determinants and relaxation
JO - Bollettino dell'Unione Matematica Italiana
DA - 2005/2//
PB - Unione Matematica Italiana
VL - 8-B
IS - 1
SP - 187
EP - 250
AB - A characterization of the total variation $TV(u, \Omega)$ of the Jacobian determinant $\det Du$ is obtained for some classes of functions $u : \Omega \rightarrow \mathbb{R}^{n}$ outside the traditional regularity space $W^{1, n}(\Omega; \mathbb{R}^{n})$. In particular, explicit formulas are deduced for functions that are locally Lipschitz continuous away from a given one point singularity $x_{0}\in \Omega$. Relations between $TV(u, \Omega)$ and the distributional determinant $\text{Det}\, Du$ are established, and an integral representation is obtained for the relaxed energy of certain polyconvex functionals at maps $u\in W^{1, p}(\Omega; \mathbb{R}^{n})\cap W^{1, \infty}(\Omega\backslash \{x_0\}; \mathbb{R}^{n})$.
LA - eng
UR - http://eudml.org/doc/194572
ER -

References

top
  1. ACERBI, E. G. - BOUCHITTÉ, G. - FONSECA, I., Relaxation of convex functionals: the gap problem, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 20 (2003), 359-390. Zbl1025.49012MR1972867
  2. ACERBI, E. - DAL MASO, G., New lower semicontinuity results for polyconvex integrals case, Calc. Var., 2 (1994), 329-372. Zbl0810.49014MR1385074
  3. ACERBI, E. - FUSCO, N., Semicontinuity problems in the calculus of variations, Arch. Rational Mech. Anal., 86 (1984), 125-145. Zbl0565.49010MR751305
  4. BALL, J. M., Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal., 63 (1977), 337-403. Zbl0368.73040MR475169
  5. BALL, J. M., Discontinuous equilibrium solutions and cavitation in nonlinear elasticity, Phil. Trans. Roy. Soc. London, A, 306 (1982), 557-611. Zbl0513.73020MR703623
  6. BETHUEL, F., A characterization of maps in H 1 ⁢ B 3 , S 2 which can be approximated by smooth maps, Ann. Inst. H. Poincaré, Anal. Non Lineaire, 7 (1990), 269-286. Zbl0708.58004MR1067776
  7. BETHUEL, F. - BREZIS, H. - HELÉIN F., F., Ginzburg-Landau Vortices, Birkhäuser, Boston, 1994. Zbl0802.35142MR1269538
  8. BOJARSKI, B. - HAJLCASZ, P., Pointwise inequalities for Sobolev functions and some applications, Studia Math., 106 (1993), 77-92. Zbl0810.46030MR1226425
  9. BOUCHITTÉ, G. - FONSECA, I. - MALÝ, J., The effective bulk energy of the relaxed energy of multiple integrals below the growth exponent, Proc. Royal Soc. Edinburgh Sect. A, 128 (1998), 463-479. Zbl0907.49008MR1632814
  10. BREZIS, H. - CORON, J. M. - LIEB, E. H., Harmonic maps with defects, Comm. Math. Phys., 107 (1986), 649-705. Zbl0608.58016MR868739
  11. BREZIS, H. - FUSCO, N. - SBORDONE, C., Integrability for the Jacobian of orientation preserving mappings, J. Funct. Anal, 115 (1993), 425-431. Zbl0847.26012MR1234399
  12. BREZIS, H. - NIRENBERG, L., Degree theory and BMO: I, Sel. Math., 2 (1995), 197-263. Zbl0852.58010MR1354598
  13. BREZIS, H. - NIRENBERG, L., Degree theory and BMO: II, Sel. Math., 3 (1996), 309-368. Zbl0868.58017MR1422201
  14. CARTAN, H., Formes différentielles, Hermann, Paris, 1967. Zbl0184.12701MR231303
  15. CELADA, P. - DAL MASO, G., Further remarks on the lower semicontinuity of polyconvex integrals, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 11 (1994), 661-691. Zbl0833.49013MR1310627
  16. CHAPMAN, S. J., A hierarchy of models for type-II superconductors, Siam Review, 42 (2000), 555-598. Zbl0967.82014MR1814048
  17. CHO, K. - GENT, A. N., Cavitation in models elastomeric composites, J. Mater. Sci., 23 (1988), 141-144. 
  18. COIFMAN, R. - LIONS, P. L. - MEYER, Y. - SEMMES, S., Compacité par compensation et espaces de Hardy, C. R. Acad. Sci. Paris, 309 (1989), 945-949. Zbl0684.46044MR1054740
  19. DACOROGNA, B., Direct Methods in Calculus of Variations, Appl. Math. Sciences, 78, Springer-Verlag, Berlin1989. Zbl0703.49001MR990890
  20. DACOROGNA, B. - MARCELLINI, P., Semicontinuité pour des intégrandes polyconvexes sans continuité des déterminants, C. R. Acad. Sci. Paris, 311 (1990), 393-396. Zbl0723.49007MR1071650
  21. DACOROGNA, B. - MURAT, F., On the optimality of certain Sobolev exponents for the weak continuity of determinants, J. Funct. Anal., 105 (1992), 42-62. Zbl0769.46025MR1156669
  22. DAL MASO, G. - SBORDONE, C., Weak lower semicontinuity of polyconvex integrals: a borderline case, Math. Z., 218 (1995), 603-609. Zbl0822.49010MR1326990
  23. FEDERER, H., Geometric measure theory, Springer-Verlag, Berlin, 1969. Zbl0874.49001MR257325
  24. FONSECA, I. - FUSCO, N. - MARCELLINI, P., On the Total Variation of the Jacobian, J. Funct. Anal., 207 (2004), 1-32. Zbl1041.49016MR2027634
  25. FONSECA, I. - GANGBO, W., Local invertibility of Sobolev functions, SIAM J. Math. Anal., 26 (1995), 280-304. Zbl0839.30018MR1320221
  26. FONSECA, I. - GANGBO, W., Degree theory in analysis and applications, Oxford Lecture Series in Mathematics and its Applications, 2. Clarendon Press, Oxford, 1995. Zbl0852.47030MR1373430
  27. FONSECA, I. - MALÝ, J., Relaxation of Multiple Integrals below the growth exponent, Anal. Inst. H. Poincaré. Anal. Non Linéaire, 14 (1997), 309-338. Zbl0868.49011MR1450951
  28. FONSECA, I. - LEONI, G. - MALÝ, J., Weak continuity of jacobian integrals. In preparation. 
  29. FONSECA, I. - MARCELLINI, P., Relaxation of multiple integrals in subcritical Sobolev spaces, J. Geometric Analysis, 7 (1997), 57-81. Zbl0915.49011MR1630777
  30. FUSCO, N. - HUTCHINSON, J. E., A direct proof for lower semicontinuity of polyconvex functionals, Manuscripta Math., 87 (1995), 35-50. Zbl0874.49015MR1329439
  31. GANGBO, W., On the weak lower semicontinuity of energies with polyconvex integrands, J. Math. Pures et Appl., 73 (1994), 455-469. Zbl0829.49011MR1300984
  32. GENT, A. N., Cavitation in rubber: a cautionary tale, Rubber Chem. Tech., 63 (1991), G49-G53. 
  33. GENT, A. N. - TOMPKINS, D. A., Surface energy effects for small holes or particles in elastomers, J. Polymer Sci., Part A, 7 (1969), 1483-1488. 
  34. GIAQUINTA, M. - MODICA, G. - SOUČEK, J., Cartesian currents, weak dipheomorphisms and existence theorems in nonlinear elasticity, Arch. Rat. Mech. Anal., 106 (1989), 97-159. Erratum and addendum: Arch. Rat. Mech. Anal., 109 (1990), 385-592. Zbl0712.73009MR980756
  35. GIAQUINTA, M. - MODICA, G. - SOUČEK, J., Cartesian Currents and Variational Problems for Mappings into Spheres, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 16 (1989), 393-485. Zbl0713.49014MR1050333
  36. GIAQUINTA, M. - MODICA, G. - SOUČEK, J., Liquid crystals: relaxed energies, dipoles, singular lines and singular points, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 17 (1990), 415-437. Zbl0760.49026MR1079984
  37. GIAQUINTA, M. - MODICA, G. - SOUČEK, J., Graphs of finite mass which cannot be approximated in area by smooth graphs, Manuscripta Math., 78 (1993), 259-271. Zbl0796.58006MR1206156
  38. GIAQUINTA, M. - MODICA, G. - SOUČEK, J., Cartesian currents in the calculus of variations I and II, Ergebnisse der Mathematik und Ihrer Grenzgebiete Vol. 38, Springer-Verlag, Berlin, 1998. Zbl0914.49002MR1645086
  39. HAJLCASZ, P., Note on weak approximation of minors, Ann. Inst. H. Poincaré, 12 (1995), 415-424. Zbl0910.49025MR1341410
  40. IWANIEC, T. - SBORDONE, C., On the integrability of the Jacobian under minimal hypotheses, Arch. Rational Mech. Anal., 119 (1992), 129-143. Zbl0766.46016MR1176362
  41. JAMES, R. D. - SPECTOR, S. J., The formation of filamentary voids in solids, J. Mech. Phys. Solids, 39 (1991), 783-813. Zbl0761.73020MR1120242
  42. JERRARD, R. L. - SONER, H. M., Functions of bounded higer variation, to appear. Zbl1057.49036MR1911049
  43. MALÝ, J., L p -approximation of Jacobians, Comment. Math. Univ. Carolin., 32 (1991), 659-666. Zbl0753.46024MR1159812
  44. MALÝ, J., Weak lower semicontinuity of polyconvex integrals, Proc. Roy. Soc. Edinburgh, 123A (1993), 681-691. Zbl0813.49017MR1237608
  45. MALÝ, J., Weak lower semicontinuity of quasiconvex integrals, Manusc. Math., 85 (1994), 419-428. Zbl0862.49017MR1305752
  46. MARCELLINI, P., Approximation of quasiconvex functions and lower semicontinuity of multiple integrals, Manuscripta Math., 51 (1985), 1-28. Zbl0573.49010MR788671
  47. MARCELLINI, P., On the definition and the lower semicontinuity of certain quasiconvex integrals, Ann. Inst. Henri Poincare, Analyse non Linéaire, 3 (1986), 391-409. Zbl0609.49009MR868523
  48. MARCELLINI, P., The stored-energy for some discontinuous deformations in nonlinear elasticity, Essays in honor of E. De Giorgi, Vol. 2, ed. F. Colombini et al., Birkhäuser, 1989, 767-786. Zbl0679.73006MR1034028
  49. MILNOR, J. M., From the differentiable viewpoint, The Univ. Press of Virginia, Charlottesville, 1965. Zbl0136.20402
  50. MORREY, C. B., Multiple integrals in the calculus of variations, Springer-Verlag, Berlin, 1966. Zbl1213.49002MR202511
  51. MÜLLER, S., Weak continuity of determinants and nonlinear elasticity, C. R. Acad. Sci. Paris, 307 (1988), 501-506. Zbl0679.34051MR964116
  52. MÜLLER, S., Det4det. A Remark on the distributional determinant, C. R. Acad. Sci. Paris, 311 (1990), 13-17. Zbl0717.46033MR1062920
  53. MÜLLER, S., Higher integrability of determinants and weak convergence in L 1 , J. Reine Angew. Math., 412 (1990), 20-34. Zbl0713.49004MR1078998
  54. MÜLLER, S., On the singular support of the distributional determinant, Annales Institut Henri Poincaré, Analyse Non Linéaire, 10 (1993), 657-696. Zbl0792.46027MR1253606
  55. MÜLLER, S. - SPECTOR, S. J., An existence theory for nonlinear elasticity that allows for cavitation, Arch. Rat. Mech. Anal., 131 (1995), 1-66. Zbl0836.73025MR1346364
  56. MÜLLER, S. - TANG, Q. - YAN, S. B., On a new class of elastic deformations not allowing for cavitation, Ann. IHP, 11 (1994), 217-243. Zbl0863.49002MR1267368
  57. MURAT, F., Compacité par compensation: condition necessaire et suffisante de continuité faible sous une hypothése de rang constant, Ann. Sc. Norm. Sup. Pisa, 8 (1981), 68-102. Zbl0464.46034MR616901
  58. RESHETNYAK, Y., Weak convergence and completely additive vector functions on a set, Sibir. Math., 9 (1968), 1039-1045. Zbl0176.44402
  59. SIVALOGANATHAN, J., Uniqueness of regular and singular equilibria for spherically symmetric problems of nonlinear elasticity, Arch. Rat. Mech. Anal., 96 (1986), 97-136. Zbl0628.73018MR853969

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.