Hessian measures. II

Neil S. Trudinger and Xu-Jia Wang

Review from Zentralblatt MATH:

In a previous paper [Topol. Methods Nonlinear Anal. 10, No. 2, 225-239 (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

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