Generalised solutions of Hessian equations

Abstract

We introduce a definition of generalised solutions of the Hessian equation S m ( D 2 u ) = f in a convex set ω ⊂ ℝ n , where S m ( D 2 u ) denotes the m -th symmetric function of the eigenvalues of D 2 u , f ∈ L p (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.

Locations

• Bulletin of the Australian Mathematical Society - View - PDF