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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 \(u\).

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