Geodesic
On weak Hessian determinants
Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 16 (2005) no. 3, pp. 159-169.

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

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. 
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D'Onofrio, Luigi; Giannetti, Flavia; Greco, Luigi. On weak Hessian determinants. Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 16 (2005) no. 3, pp. 159-169. http://geodesic.mathdoc.fr/item/RLIN_2005_9_16_3_a1/

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

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

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

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

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

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

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

[8] L. Greco, A remark on the equality $\text{det} Df = \text{Det} Df$. Differential Integral Equations, 6, n. 5, 1993, 1089-1100. | MR | Zbl

[9] C.E. Gutiérrez, The Monge-Ampère Equation. Birkhäuser, Boston 2001. | Zbl

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

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

[12] J. Malý, 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

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

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

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

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

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

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