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Annals of Mathematics, II. Series Vol. 150, No. 2, pp. 579604 (1999) 

Hessian measures. IINeil S. Trudinger and XuJia WangReview from Zentralblatt MATH: In a previous paper [Topol. Methods Nonlinear Anal. 10, No. 2, 225239 (1997; Zbl 0915.35039)] the same authors introduced the notion of $k$Hessian measures associated with a continuous $k$convex function in a domain $\Omega\subset \bbfR^n$, $k=1, \dots, n$, and proved a weak continuity result with respect to local uniform convergence. In the present paper they consider upper semicontinuous $k$convex functions and prove weak continuity of the corresponding $k$Hessian measure with respect to convergence in measure. To get this result, they first prove some lemmas and theorems for $k$convex functions which may have own interest. Then, some local integral estimates for the $k$Hessian operator $F_k[u]$ and for the gradient $Du$ in terms of the integral of $u$ are proved. Using the above results, the following interesting theorem is proved: For any $k$convex function $u$, there exists a Borel measure $\mu_k[u]$ in $\Omega$ such that: (i) $\mu_k[u]=F_k[u]$ for $u\in C^2(\Omega)$, and (ii) if $\{u_m\}$ is a sequence of $k$convex functions converging locally in measure to a $k$convex function $u$, the sequence of Borel measures $\{\mu_k[u]\}$ converges weakly to $\mu_k[u]$. Reviewed by G.Porru Keywords: $k$Hessian measures; $k$convex functions Classification (MSC2000): 35J60 28A33 35B05 31B15 Full text of the article:
Electronic fulltext finalized on: 8 Sep 2001. This page was last modified: 21 Jan 2002.
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