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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
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Abstract

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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

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