Duke Mathematical Journal

Singularities and rank one properties of Hessian measures

Patricio Aviles and Yoshikazu Giga

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Duke Math. J., Volume 58, Number 2 (1989), 441-467.

First available in Project Euclid: 20 February 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 49Q15: Geometric measure and integration theory, integral and normal currents [See also 28A75, 32C30, 58A25, 58C35]
Secondary: 26B12: Calculus of vector functions 28A78: Hausdorff and packing measures


Aviles, Patricio; Giga, Yoshikazu. Singularities and rank one properties of Hessian measures. Duke Math. J. 58 (1989), no. 2, 441--467. doi:10.1215/S0012-7094-89-05820-1. https://projecteuclid.org/euclid.dmj/1077307533

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  • [1] V. I. Arnold, Singularities of systems of rays, Russian Math. Surveys 38 (1983), no. 2, 87–176, transl. of Uspehi M. N.
  • [2] P. Aviles and Y. Giga, A mathematical problem related to the physical theory of liquid crystal configurations, Miniconference on geometry and partial differential equations, 2 (Canberra, 1986) eds. L. Simon and J. Hutchinson, Proc. Centre Math. Anal. Austral. Nat. Univ., vol. 12, Austral. Nat. Univ., Canberra, 1987, pp. 1–16.
  • [3] G. Dal Maso, Integral representation on $\rm BV(\Omega )$ of $\Gamma$-limits of variational integrals, Manuscripta Math. 30 (1979/80), no. 4, 387–416.
  • [4] E. De Giorgi, Nuovi teoremi relativi alle misure $(r-1)$-dimensionali in uno spazio ad $r$ dimensioni, Ricerche Mat. 4 (1955), 95–113.
  • [5] E. De Giorgi, L. Ambrosio, and G. Buttazzo, Integral representation and relaxation for functionals defined on measures, Preprint.
  • [6] F. Demengel, Fonctions à hessien borné, Ann. Inst. Fourier (Grenoble) 34 (1984), no. 2, 155–190.
  • [7] F. Demengel and R. Temam, Convex functions of a measure and applications, Indiana Univ. Math. J. 33 (1984), no. 5, 673–709.
  • [8] R. J. DiPerna, Singularities of solutions of nonlinear hyperbolic systems of conservation laws, Arch. Rational Mech. Anal. 60 (1975/76), no. 1, 75–100.
  • [9] R. J. DiPerna, The structure of solutions to hyperbolic conservation laws, Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. IV ed. R. J. Knops, Res. Notes in Math., vol. 39, Pitman, Boston, Mass., 1979, pp. 1–16.
  • [10]1 R. M. Dudley, On second derivatives of convex functions, Math. Scand. 41 (1977), no. 1, 159–174.
  • [10]2 R. M. Dudley, Acknowledgment of priority: “On second derivatives of convex functions”, Math. Scand. 46 (1980), no. 1, 61.
  • [11] C. L. Evans and R. Gariepy, Lectures on Geometric Measure Theory, to appear.
  • [12] H. Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969.
  • [13] W. H. Fleming, The Cauchy problem for a nonlinear first order partial differential equation, J. Differential Equations 5 (1969), 515–530.
  • [14] E. Giusti, Minimal surfaces and functions of bounded variation, Monographs in Mathematics, vol. 80, Birkhäuser Verlag, Basel, 1984.
  • [15] C. Goffman and J. Serrin, Sublinear functions of measures and variational integrals, Duke Math. J. 31 (1964), 159–178.
  • [16] P. L. Lions, Generalized solutions of Hamilton-Jacobi equations, Research Notes in Mathematics, vol. 69, Pitman (Advanced Publishing Program), Boston, Mass., 1982.
  • [17] L. Simon, Lectures on geometric measure theory, Proceedings of the Centre for Mathematical Analysis, Australian National University, vol. 3, Australian National University Centre for Mathematical Analysis, Canberra, 1983.
  • [18] R. Temam, Approximation de fonctions convexes sur un espace de mesures et applications, Canad. Math. Bull. 25 (1982), no. 4, 392–413.
  • [19] A. I. Vol'pert, The Spaces BV and quasilinear equations, Math. USSR-Sbornik 2 (1967), 255–267, English translation.