A derivation theorem for capacities with respect to a Radon measure
Let \(\mu\) be a Borel measure and let \(\nu\) be a Radon measure such that \(\mu\) is absolutely continuous with respect to \(\nu\). Then it is a classical result that \(\frac{d\mu}{d\nu}(x)=\lim_{r\to 0}\frac{\mu (B_ r(x))}{\nu (B_ r(x))}\nu\)-almost everywhere; i.e. the Radon-Nikodým derivative can be identified by the values of the two measures on balls. However the values of \(\mu\) on balls are not available in some problems of Calculus of Variations where there are only informations of variational nature on the measure \(\mu\). In this paper the authors prove an alternative result on the computation of \(d\mu /d\nu\) which makes use of the concept of \(\mu\)-capacity of a set and therefore it can be easily employed in the above mentioned situations. In fact they prove that, if \(\mu\) and \(\nu\) are zero on every set of capacity zero, then \[ \frac{d\mu}{d\nu}(x)=\lim_{r\to 0}\frac{cap_{\mu}(B_ r(x),B_{2r}(x))}{\nu (B_ r(x))} \] \(\nu\)-almost everywhere, where \[ cap_{\mu}(B_ r(x),B_{2r}(x))=\min \{\int_{B_{2r}}| Du|^ 2dx+\int_{B_ r}u^ 2d\mu: u-1\in H^ 1_ 0(B_{2r})\}. \] Moreover they use this result in order to characterize the limit of a variational convergence of measure introduced in \textit{G. Dal Maso} and \textit{U. Mosco} [Appl. Math. Optimization 15, 15-64 (1987)].
limit of a variational convergence of measure
scientific article; zbMATH DE number 4008728