Regularization of a set function -- application to integral representation (Q1864705)

In this paper the authors consider a variational functional \({\mathcal F}(u;A)\) assuming that the gauge function is majorized by a given random measure \(\mu\) on \({\mathbb{R}}^n\). In the context of integral representation \(\mu\) is associated with the growth condition. The main result is that for a fixed \(\mu\), the measure \(\lambda:={\mathcal F}(u;\cdot)\) can be recovered from the set function \(S:=m(u;\cdot)\) due to a variant of the Carathéodory construction, where \[ m(u;A)=\inf \{{\mathcal F}(u+\varphi;A):\operatorname{spt}\varphi\subset\subset A\}. \]