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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
##### Keywords:
Hessian operator; $$k$$-convex functions
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