zbMATH — the first resource for mathematics

Hessian measures. I. (English) Zbl 0915.35039
Let \(\Omega\) be a domain in Euclidean \(n\)-space \(\mathbb{R}^n\). For \(k=1,\dots, n\) and \(u\in C^2(\Omega)\) the \(k\)-Hessian operator \(F_k\) is defined by \(F_k[u]= S_k(\lambda(D^2u))\), where \(\lambda= (\lambda_1,\dots, \lambda_n)\) denotes the eigenvalues of the Hessian matrix of second derivatives \(D^2u\), and \(S_k\) is the \(k\)th elementary symmetric function on \(\mathbb{R}^n\), given by \[ S_k(\lambda)= \sum_{i_1<\cdots<i_k} \lambda_{i_1}\cdots\lambda_{i_k}. \] Our purpose in this paper is to extend the definition of the \(F_k\) to corresponding classes of continuous functions so that \(F_k[u]\) is a Borel measure and to consider the Dirichlet problem in this setting. We shall prove that \(F_k[u]\) may be extended to the class of \(k\)-convex functions in \(C^0(\Omega)\) as a Borel measure \(\mu_k\), for all \(k= 1,\dots,n,\), and that the corresponding mapping \(u\to \mu_k[u]\) is weakly continuous on \(C^0(\Omega)\). The resultant measure \(\mu_k[u]\) will be called the \(k\)-Hessian measure generated by \(u\).

35J60 Nonlinear elliptic equations
28A33 Spaces of measures, convergence of measures
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
Full Text: DOI