On weak Hessian determinants
D'Onofrio, Luigi ; Giannetti, Flavia ; Greco, Luigi
Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Tome 16 (2005), p. 159-169 / Harvested from Biblioteca Digitale Italiana di Matematica
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We consider and study several weak formulations of the Hessian determinant, arising by formal integration by parts. Our main concern are their continuity properties. We also compare them with the Hessian measure.

Consideriamo ed esaminiamo varie formulazioni deboli del determinante hessiano, definite come distribuzioni di Schwartz mediante integrazione per parti, principalmente riguardo alle loro proprietà di continuità. Confrontiamo inoltre tali formulazioni deboli con la misura hessiana.

Publié le : 2005-09-01
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     author = {Luigi D'Onofrio and Flavia Giannetti and Luigi Greco},
     title = {On weak Hessian determinants},
     journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
     volume = {16},
     year = {2005},
     pages = {159-169},
     zbl = {1127.26007},
     mrnumber = {2227740},
     language = {en},
     url = {http://dml.mathdoc.fr/item/RLIN_2005_9_16_3_159_0}
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D'Onofrio, Luigi; Giannetti, Flavia; Greco, Luigi. On weak Hessian determinants. Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Tome 16 (2005) pp. 159-169. http://gdmltest.u-ga.fr/item/RLIN_2005_9_16_3_159_0/

[1] Adams, R.A., Sobolev spaces. Pure and Applied Mathematics, vol. 65, Academic Press, 1975. | MR 450957 | Zbl 0314.46030

[2] Ball, J.M., Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal., 63, n. 4, 1976-77, 337-403. | MR 475169 | Zbl 0368.73040

[3] Ball, J.M. - Curie, J.C. - Olver, P.J., Null Lagrangians, weak continuity, and variational problems of arbitrary order. J. Funct. Anal., 41, 1981, 135-174. | MR 615159 | Zbl 0459.35020

[4] Coifman, R. - Lions, P.-L. - Meyer, Y. - Semmes, S., Compensated compactness and Hardy spaces. J. Math. Pures Appl., 72, 1993, 247-286. | MR 1225511 | Zbl 0864.42009

[5] Dacorogna, B. - Murat, F., On the optimality of certain Sobolev exponents for the weak continuity of determinants. J. Funct. Anal., 105, n. 1, 1992, 42-62. | MR 1156669 | Zbl 0769.46025

[6] Evans, L.C. - Gariepy, R.F., Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton, FL 1992. | MR 1158660 | Zbl 0804.28001

[7] Giaquinta, M. - Modica, G. - Souček, J., Area and the area formula. Rend. Sem. Mat. Fis. Milano, 62, 1992, 53-87. | MR 1293774 | Zbl 0828.49022

[8] Greco, L., A remark on the equality det⁢D⁢f=Det⁢D⁢f. Differential Integral Equations, 6, n. 5, 1993, 1089-1100. | MR 1230483 | Zbl 0784.49013

[9] Gutiérrez, C.E., The Monge-Ampère Equation. Birkhäuser, Boston2001. | Zbl 0989.35052

[10] Iwaniec, T., On the concept of the weak Jacobian and Hessian. Report. Univ. Jyväskylä, 83, 2001, 181-205. | MR 1886622 | Zbl 1007.35015

[11] Iwaniec, T. - Martin, G., Geometric Function Theory and Nonlinear Analysis. Oxford University Press, Oxford 2001. | MR 1859913 | Zbl 1045.30011

[12] Malý, J., From Jacobians to Hessian: distributional form and relaxation. In: Proceedings of the Trends in the Calculus of Variations (Parma, September 15-18, 2004). Riv. Mat. Univ. Parma, in press; http:// www.math.cmu.edu/$ nw0z/publications/05-CNA-003/003abs/05-CNA-003.pdf. | Zbl 1101.49011

[13] Miranda, C., Su alcuni teoremi di inclusione. Annales Polonici Math., 16, 1965, 305-315. | MR 187077 | Zbl 0172.40303

[14] Müller, S., Det = det, a remark on the distributional determinant. C.R. Acad. Sci. Paris, Sér. I Math., 311, 1990, 13-17. | MR 1062920 | Zbl 0717.46033

[15] Müller, S., On the singular support of the distributional determinant. Ann. Inst. H. Poincaré Anal. Non Linéaire, 10, 1993, 657-698. | MR 1253606 | Zbl 0792.46027

[16] Nirenberg, L., An extended interpolation inequality. Ann. Sc. Norm. Sup. Pisa Sci. fis. mat., 20, 1966, 733-737. | MR 208360 | Zbl 0163.29905

[17] Olver, P.J., Hyper-Jacobians, determinantal ideals and weak solutions to variational problems. Proc. Roy. Soc. Edinburgh Sect. A, 95, 1983, n. 3-4, 317-340. | MR 726882 | Zbl 0562.49005

[18] Trudinger, N.G., Weak solutions of Hessian equations. Comm. Partial Differential Equations, 22, 1997, n. 7-8, 1251-1261. | MR 1466315 | Zbl 0883.35035