Hostname: page-component-745bb68f8f-lrblm Total loading time: 0 Render date: 2025-01-07T16:50:36.188Z Has data issue: false hasContentIssue false
  • English
  • Français

Generalised solutions of Hessian equations

Published online by Cambridge University Press:  17 April 2009

Andrea Colesanti  and
Paolo Salani
Andrea Colesanti
Affiliation:
Dipartimento di Matematica U DiniVialie Morgagni 67/AFirenzeItaly
Paolo Salani
Affiliation:
Dipartimento di Matematica U DiniVialie Morgagni 67/AFirenzeItaly
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We introduce a definition of generalised solutions of the Hessian equation Sm(D2u) = f in a convex set ω ⊂ ℝn, where Sm(D2u) denotes the m-th symmetric function of the eigenvalues of D2u, f ∈ Lp(ω), p ≥ 1, and m ∈ {1, …, n}. Such a definition is given in the class of semi-convex functions, and it extends the definition of convex generalised solutions for the Monge-Ampère equation. We prove that semiconvex weak solutions are solutions in the sense of the present paper.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 56 , Issue 3 , December 1997 , pp. 459 - 466
Copyright
Copyright © Australian Mathematical Society 1997

References

[1]Ash, R.B., Measure, integration and functional analysis (Academic Press, New York, 1972).Google Scholar
[2]Bakelman, I.J., ‘Geometric inequalities and existence theorems for convex generalized solutions of n-dimensional Monge-Ampére equations’, in Geometric analysis and nonlinear partial differential equations, (Bakelman, I.J., Editor) (Marcel Dekker, New York, 1993), pp. 237–287.Google Scholar
[3]Bangert, V., ‘Sets with positive reach’, Arch. Math. 38 (1982), 54–57.CrossRefGoogle Scholar
[4]Colesanti, A., ‘A Steiner type formula for convex functions’, Mathematika (to appear).Google Scholar
[5]Caffarelli, L., Nirenberg, L. and Spruck, J., ‘The Dirichlet problem for nonlinear second order elliptic equations, III: functions of the eigenvalues of the Hessian’, Acta Math. 155 (1985), 261–301.CrossRefGoogle Scholar
[6]Federer, H., ‘Curvature measures’, Trans. Amer. Math. Soc. 93 (1959), 418–491.CrossRefGoogle Scholar
[7]Fu, J.H.G., ‘Tubular neighborhoods in Euclidean spaces’, Duke Math J. 52 (1985), 1025–1046.CrossRefGoogle Scholar
[8]Rauch, J. and Taylor, B.A., ‘The Dirichlet problem for the multidimensional Monge-Ampere equation’, Rocky Mountain J. Math. 7 (1977), 345–364.CrossRefGoogle Scholar
[9]Schneider, R., Convex bodies: the Brunn-Minkowski theory (University Press, Cambridge, 1993).CrossRefGoogle Scholar
[10]Trudinger, N.S., ‘On the Dirichlet problem for Hessian equations’, Acta Math. 175 (1995), 151–164.CrossRefGoogle Scholar
[11]Trudinger, N.S., ‘Weak Solutions of Hessian equations’, (preprint).Google Scholar
[12]Urbas, J.I.E., ‘Regularity of generalized solutions of Monge-Ampère equations’, Math. Z. 197 (1988), 365–393.CrossRefGoogle Scholar