an:01419958
Zbl 0947.35055
Trudinger, Neil S.; Wang, Xu-Jia
Hessian measures. II
EN
Ann. Math. (2) 150, No. 2, 579-604 (1999).
00061059
1999
j
35J60 28A33 35B05 31B15
\(k\)-Hessian measures; \(k\)-convex functions
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 \(\Omega\subset \mathbb{R}^n\), \(k=1, \dots, n\), 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 \(\mu_k[u]\).
G.Porru (Cagliari)
Zbl 0915.35039