Singularities and rank one properties of Hessian measures (Q923361)

This paper is concerned with an arbitrary locally integrable function, u, (defined on an open subset of \({\mathbb{R}}^ N)\) whose second partial derivatives, \(\mu_{ij}\), are Radon measures. Let [\(\cdot]\) denote a norm on the space of \(N\times N\) real matrices, and let [\(\mu\) ] be the variation measure corresponding to that norm and to the matrix of measures \((\mu_{ij})\). The authors define the singular set of the measure [\(\mu\) ] to consist of those points where the lower N-dimensional density of [\(\mu\) ] is infinite. The authors' main result is that on a non-trivial subset of that singular set the matrix \((\mu_{ij})\) has rank one. More precisely, the authors let \(S^*\) be the set of points x such that \[ A_{ij}(x)=\lim_{r\downarrow 0}\frac{\mu_{ij}(B_ r(x))}{[\mu](B_ r(x))}\quad exists\text{ for } 1\leq i,j\leq N, \] \[ | A_{ij}(x)| =\lim_{r\downarrow 0}\frac{| \mu_{ij}| (B_ r(x))}{[\mu](B_ r(x))}\text{ for } 1\leq i,j\leq N, \] \[ [A(x)]=1,\quad \Theta_*^ N([\mu],x)=\infty. \] The matrix \(A=(A_{ij})\) may then be thought of as the second derivative of u at points of \(S^*\). Their result (Corollary to Theorem A) is that if \(K\subset S^*\) and the q-dimensional Hausdorff measure of K is finite for some \(q<N\), then A has rank one of K, up to a set of q-dimensional Hausdorff measure zero.