Topological degree, Jacobian determinants and relaxation

Fonseca, Irene; Fusco, Nicola; Marcellini, Paolo

Fonseca, Irene ; Fusco, Nicola ; Marcellini, Paolo

Bollettino dell'Unione Matematica Italiana, Tome 8-A (2005), p. 187-250 / Harvested from Biblioteca Digitale Italiana di Matematica

Access to full text
Full (PDF)
Full (DjVu)

Abstract
Résumé

A characterization of the total variation $T V (u, Ω)$ of the Jacobian determinant $det D u$ is obtained for some classes of functions $u : Ω \to R^{n}$ 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 $x_{0} \in Ω$ . 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 $u \in W^{1, p} (Ω; R^{n}) \cap W^{1, \infty} (Ω \ \{x_{0}\}; R^{n})$ .

Si ottiene una caratterizzazione della variazione totale $T V (u, Ω)$ del determinante Jacobiano $det D u$ per alcune classi di applicazioni $u : Ω \to R^{n}$ che non fanno parte della tradizionale classe di Sobolev $W^{1, n} (Ω; R^{n})$ . In particolare, si forniscono formule esplicite per applicazioni localmente Lipschitziane al di fuori di un punto isolato $x_{0} \in Ω$ . Si stabiliscono anche alcune relazioni fra $T V (u, Ω)$ e il determinante distribuzionale $Det D u$ . Inoltre si fornisce una rappresentazione integrale per l'energia rilassata di certi integrali policonvessi relativi ad applicazioni $u \in W^{1, p} (Ω; R^{n}) \cap W^{1, \infty} (Ω ∖ \{x_{0}\}; R^{n})$ .

Publié le : 2005-02-01

MR 2122983 | Zbl 1177.49066

@article{BUMI_2005_8_8B_1_187_0,
     author = {Irene Fonseca and Nicola Fusco and Paolo Marcellini},
     title = {Topological degree, Jacobian determinants and relaxation},
     journal = {Bollettino dell'Unione Matematica Italiana},
     volume = {8-A},
     year = {2005},
     pages = {187-250},
     zbl = {1177.49066},
     mrnumber = {2122983},
     language = {en},
     url = {http://dml.mathdoc.fr/item/BUMI_2005_8_8B_1_187_0}
}

Fonseca, Irene; Fusco, Nicola; Marcellini, Paolo. Topological degree, Jacobian determinants and relaxation. Bollettino dell'Unione Matematica Italiana, Tome 8-A (2005) pp. 187-250. http://gdmltest.u-ga.fr/item/BUMI_2005_8_8B_1_187_0/

Bibliographie

[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. | MR 1972867 | Zbl 1025.49012

[2] Acerbi, E. - Dal Maso, G., New lower semicontinuity results for polyconvex integrals case, Calc. Var., 2 (1994), 329-372. | MR 1385074 | Zbl 0810.49014

[3] Acerbi, E. - Fusco, N., Semicontinuity problems in the calculus of variations, Arch. Rational Mech. Anal., 86 (1984), 125-145. | MR 751305 | Zbl 0565.49010

[4] Ball, J. M., Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal., 63 (1977), 337-403. | MR 475169 | Zbl 0368.73040

[5] Ball, J. M., Discontinuous equilibrium solutions and cavitation in nonlinear elasticity, Phil. Trans. Roy. Soc. London, A, 306 (1982), 557-611. | MR 703623 | Zbl 0513.73020

[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. | MR 1067776 | Zbl 0708.58004

[7] Bethuel, F. - Brezis, H. - Heléin F., F., Ginzburg-Landau Vortices, Birkhäuser, Boston, 1994. | MR 1269538 | Zbl 0802.35142

[8] Bojarski, B. - Hajlcasz, P., Pointwise inequalities for Sobolev functions and some applications, Studia Math., 106 (1993), 77-92. | MR 1226425 | Zbl 0810.46030

[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. | MR 1632814 | Zbl 0907.49008

[10] Brezis, H. - Coron, J. M. - Lieb, E. H., Harmonic maps with defects, Comm. Math. Phys., 107 (1986), 649-705. | MR 868739 | Zbl 0608.58016

[11] Brezis, H. - Fusco, N. - Sbordone, C., Integrability for the Jacobian of orientation preserving mappings, J. Funct. Anal, 115 (1993), 425-431. | MR 1234399 | Zbl 0847.26012

[12] Brezis, H. - Nirenberg, L., Degree theory and BMO: I, Sel. Math., 2 (1995), 197-263. | MR 1354598 | Zbl 0852.58010

[13] Brezis, H. - Nirenberg, L., Degree theory and BMO: II, Sel. Math., 3 (1996), 309-368. | MR 1422201 | Zbl 0868.58017

[14] Cartan, H., Formes différentielles, Hermann, Paris, 1967. | MR 231303 | Zbl 0184.12701

[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. | MR 1310627 | Zbl 0833.49013

[16] Chapman, S. J., A hierarchy of models for type-II superconductors, Siam Review, 42 (2000), 555-598. | MR 1814048 | Zbl 0967.82014

[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. | MR 1054740 | Zbl 0684.46044

[19] Dacorogna, B., Direct Methods in Calculus of Variations, Appl. Math. Sciences, 78, Springer-Verlag, Berlin1989. | MR 990890 | Zbl 0703.49001

[20] Dacorogna, B. - Marcellini, P., Semicontinuité pour des intégrandes polyconvexes sans continuité des déterminants, C. R. Acad. Sci. Paris, 311 (1990), 393-396. | MR 1071650 | Zbl 0723.49007

[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. | MR 1156669 | Zbl 0769.46025

[22] Dal Maso, G. - Sbordone, C., Weak lower semicontinuity of polyconvex integrals: a borderline case, Math. Z., 218 (1995), 603-609. | MR 1326990 | Zbl 0822.49010

[23] Federer, H., Geometric measure theory, Springer-Verlag, Berlin, 1969. | MR 257325 | Zbl 0874.49001

[24] Fonseca, I. - Fusco, N. - Marcellini, P., On the Total Variation of the Jacobian, J. Funct. Anal., 207 (2004), 1-32. | MR 2027634 | Zbl 1041.49016

[25] Fonseca, I. - Gangbo, W., Local invertibility of Sobolev functions, SIAM J. Math. Anal., 26 (1995), 280-304. | MR 1320221 | Zbl 0839.30018

[26] Fonseca, I. - Gangbo, W., Degree theory in analysis and applications, Oxford Lecture Series in Mathematics and its Applications, 2. Clarendon Press, Oxford, 1995. | MR 1373430 | Zbl 0852.47030

[27] Fonseca, I. - Malý, J., Relaxation of Multiple Integrals below the growth exponent, Anal. Inst. H. Poincaré. Anal. Non Linéaire, 14 (1997), 309-338. | MR 1450951 | Zbl 0868.49011

[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. | MR 1630777 | Zbl 0915.49011

[30] Fusco, N. - Hutchinson, J. E., A direct proof for lower semicontinuity of polyconvex functionals, Manuscripta Math., 87 (1995), 35-50. | MR 1329439 | Zbl 0874.49015

[31] Gangbo, W., On the weak lower semicontinuity of energies with polyconvex integrands, J. Math. Pures et Appl., 73 (1994), 455-469. | MR 1300984 | Zbl 0829.49011

[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. | MR 980756 | Zbl 0712.73009

[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. | MR 1050333 | Zbl 0713.49014

[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. | MR 1079984 | Zbl 0760.49026

[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. | MR 1206156 | Zbl 0796.58006

[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. | MR 1645086 | Zbl 0914.49002

[39] Hajlcasz, P., Note on weak approximation of minors, Ann. Inst. H. Poincaré, 12 (1995), 415-424. | MR 1341410 | Zbl 0910.49025

[40] Iwaniec, T. - Sbordone, C., On the integrability of the Jacobian under minimal hypotheses, Arch. Rational Mech. Anal., 119 (1992), 129-143. | MR 1176362 | Zbl 0766.46016

[41] James, R. D. - Spector, S. J., The formation of filamentary voids in solids, J. Mech. Phys. Solids, 39 (1991), 783-813. | MR 1120242 | Zbl 0761.73020

[42] Jerrard, R. L. - Soner, H. M., Functions of bounded higer variation, to appear. | MR 1911049 | Zbl 1057.49036

[43] Malý, J., $L^{p}$ -approximation of Jacobians, Comment. Math. Univ. Carolin., 32 (1991), 659-666. | MR 1159812 | Zbl 0753.46024

[44] Malý, J., Weak lower semicontinuity of polyconvex integrals, Proc. Roy. Soc. Edinburgh, 123A (1993), 681-691. | MR 1237608 | Zbl 0813.49017

[45] Malý, J., Weak lower semicontinuity of quasiconvex integrals, Manusc. Math., 85 (1994), 419-428. | MR 1305752 | Zbl 0862.49017

[46] Marcellini, P., Approximation of quasiconvex functions and lower semicontinuity of multiple integrals, Manuscripta Math., 51 (1985), 1-28. | MR 788671 | Zbl 0573.49010

[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. | MR 868523 | Zbl 0609.49009

[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. | MR 1034028 | Zbl 0679.73006

[49] Milnor, J. M., From the differentiable viewpoint, The Univ. Press of Virginia, Charlottesville, 1965. | Zbl 0136.20402

[50] Morrey, C. B., Multiple integrals in the calculus of variations, Springer-Verlag, Berlin, 1966. | MR 202511 | Zbl 1213.49002

[51] Müller, S., Weak continuity of determinants and nonlinear elasticity, C. R. Acad. Sci. Paris, 307 (1988), 501-506. | MR 964116 | Zbl 0679.34051

[52] Müller, S., Det4det. A Remark on the distributional determinant, C. R. Acad. Sci. Paris, 311 (1990), 13-17. | MR 1062920 | Zbl 0717.46033

[53] Müller, S., Higher integrability of determinants and weak convergence in $L^{1}$ , J. Reine Angew. Math., 412 (1990), 20-34. | MR 1078998 | Zbl 0713.49004

[54] Müller, S., On the singular support of the distributional determinant, Annales Institut Henri Poincaré, Analyse Non Linéaire, 10 (1993), 657-696. | MR 1253606 | Zbl 0792.46027

[55] Müller, S. - Spector, S. J., An existence theory for nonlinear elasticity that allows for cavitation, Arch. Rat. Mech. Anal., 131 (1995), 1-66. | MR 1346364 | Zbl 0836.73025

[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. | MR 1267368 | Zbl 0863.49002

[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. | MR 616901 | Zbl 0464.46034

[58] Reshetnyak, Y., Weak convergence and completely additive vector functions on a set, Sibir. Math., 9 (1968), 1039-1045. | Zbl 0176.44402

[59] Sivaloganathan, J., Uniqueness of regular and singular equilibria for spherically symmetric problems of nonlinear elasticity, Arch. Rat. Mech. Anal., 96 (1986), 97-136. | MR 853969 | Zbl 0628.73018