Geodesic
Topological degree, Jacobian determinants and relaxation
Bollettino della Unione matematica italiana, Série 8, 8B (2005) no. 1, pp. 187-250.

Voir la notice de l'article provenant de la source Biblioteca Digitale Italiana di Matematica

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 $u\in W^{1, p}(\Omega; \mathbb{R}^{n})\cap W^{1, \infty}(\Omega\backslash \{x_0\}; \mathbb{R}^{n})$.
Si ottiene una caratterizzazione della variazione totale $TV(u, \Omega)$ del determinante Jacobiano $\det Du$ per alcune classi di applicazioni $u : \Omega \rightarrow \mathbb{R}^{n}$ che non fanno parte della tradizionale classe di Sobolev $W^{1, n}(\Omega; \mathbb{R}^{n})$. In particolare, si forniscono formule esplicite per applicazioni localmente Lipschitziane al di fuori di un punto isolato $x_{0}\in \Omega$. Si stabiliscono anche alcune relazioni fra $TV(u, \Omega)$ e il determinante distribuzionale $\text{Det}\, Du$. Inoltre si fornisce una rappresentazione integrale per l'energia rilassata di certi integrali policonvessi relativi ad applicazioni $u\in W^{1, p}(\Omega; \mathbb{R}^{n})\cap W^{1, \infty}(\Omega\setminus \{x_0\}; \mathbb{R}^{n})$. 
@article{BUMI_2005_8_8B_1_a10,
     author = {Fonseca, Irene and Fusco, Nicola and Marcellini, Paolo},
     title = {Topological degree, {Jacobian} determinants and relaxation},
     journal = {Bollettino della Unione matematica italiana},
     pages = {187--250},
     publisher = {mathdoc},
     volume = {Ser. 8, 8B},
     number = {1},
     year = {2005},
     zbl = {1177.49066},
     mrnumber = {1972867},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/BUMI_2005_8_8B_1_a10/}
}
TY  - JOUR
AU  - Fonseca, Irene
AU  - Fusco, Nicola
AU  - Marcellini, Paolo
TI  - Topological degree, Jacobian determinants and relaxation
JO  - Bollettino della Unione matematica italiana
PY  - 2005
SP  - 187
EP  - 250
VL  - 8B
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/BUMI_2005_8_8B_1_a10/
LA  - en
ID  - BUMI_2005_8_8B_1_a10
ER  - 
%0 Journal Article
%A Fonseca, Irene
%A Fusco, Nicola
%A Marcellini, Paolo
%T Topological degree, Jacobian determinants and relaxation
%J Bollettino della Unione matematica italiana
%D 2005
%P 187-250
%V 8B
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/BUMI_2005_8_8B_1_a10/
%G en
%F BUMI_2005_8_8B_1_a10
Fonseca, Irene; Fusco, Nicola; Marcellini, Paolo. Topological degree, Jacobian determinants and relaxation. Bollettino della Unione matematica italiana, Série 8, 8B (2005) no. 1, pp. 187-250. http://geodesic.mathdoc.fr/item/BUMI_2005_8_8B_1_a10/

[1] E. G. Acerbi - G. Bouchitté - I. Fonseca, Relaxation of convex functionals: the gap problem, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 20 (2003), 359-390. | fulltext mini-dml | MR | Zbl

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

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

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

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

[6] F. Bethuel, 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. | fulltext mini-dml | MR | Zbl

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

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

[9] G. Bouchitté - I. Fonseca - J. Malý, 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 | Zbl

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

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

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

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

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

[15] P. Celada - G. Dal Maso, Further remarks on the lower semicontinuity of polyconvex integrals, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 11 (1994), 661-691. | fulltext mini-dml | MR | Zbl

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

[17] K. Cho - A. N. Gent, Cavitation in models elastomeric composites, J. Mater. Sci., 23 (1988), 141-144.

[18] R. Coifman - P. L. Lions - Y. Meyer - S. Semmes, Compacité par compensation et espaces de Hardy, C. R. Acad. Sci. Paris, 309 (1989), 945-949. | fulltext mini-dml | MR | Zbl

[19] B. Dacorogna, Direct Methods in Calculus of Variations, Appl. Math. Sciences, 78, Springer-Verlag, Berlin 1989. | MR | Zbl

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

[21] B. Dacorogna - F. Murat, On the optimality of certain Sobolev exponents for the weak continuity of determinants, J. Funct. Anal., 105 (1992), 42-62. | MR | Zbl

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

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

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

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

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

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

[28] I. Fonseca - G. Leoni - J. Malý, Weak continuity of jacobian integrals. In preparation.

[29] I. Fonseca - P. Marcellini, Relaxation of multiple integrals in subcritical Sobolev spaces, J. Geometric Analysis, 7 (1997), 57-81. | MR | Zbl

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

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

[32] A. N. Gent, Cavitation in rubber: a cautionary tale, Rubber Chem. Tech., 63 (1991), G49-G53.

[33] A. N. Gent - D. A. Tompkins, Surface energy effects for small holes or particles in elastomers, J. Polymer Sci., Part A, 7 (1969), 1483-1488.

[34] M. Giaquinta - G. Modica - J. Souček, 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 | Zbl

[35] M. Giaquinta - G. Modica - J. Souček, Cartesian Currents and Variational Problems for Mappings into Spheres, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 16 (1989), 393-485. | fulltext mini-dml | MR | Zbl

[36] M. Giaquinta - G. Modica - J. Souček, Liquid crystals: relaxed energies, dipoles, singular lines and singular points, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 17 (1990), 415-437. | fulltext mini-dml | MR | Zbl

[37] M. Giaquinta - G. Modica - J. Souček, Graphs of finite mass which cannot be approximated in area by smooth graphs, Manuscripta Math., 78 (1993), 259-271. | MR | Zbl

[38] M. Giaquinta - G. Modica - J. Souček, Cartesian currents in the calculus of variations I and II, Ergebnisse der Mathematik und Ihrer Grenzgebiete Vol. 38, Springer-Verlag, Berlin, 1998. | MR | Zbl

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

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

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

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

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

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

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

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

[47] P. Marcellini, On the definition and the lower semicontinuity of certain quasiconvex integrals, Ann. Inst. Henri Poincare, Analyse non Linéaire, 3 (1986), 391-409. | fulltext mini-dml | MR | Zbl

[48] P. Marcellini, 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 | Zbl

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

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

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

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

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

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

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

[56] S. Müller - Q. Tang - S. B. Yan, On a new class of elastic deformations not allowing for cavitation, Ann. IHP, 11 (1994), 217-243. | fulltext mini-dml | MR | Zbl

[57] F. Murat, 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. | fulltext mini-dml | MR | Zbl

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

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