Local mappings on spaces of differentiable functions
The authors prove that a functional \(F(u,B)\) defined for every \(C^ k\)- function \(u\in C^ k(\Omega;\mathbb{R}^ m)\) (for \(\Omega\) a bounded open subset in \(\mathbb{R}^ n\)) and every Borel subset \(B\) of \(\Omega\) under very mild assumptions admits an integral representation \[ F(u,B)= \int_ B f(x, D^ k u(x)) d\mu(x) \] for a suitable measure \(\mu\). Such integral representations are very important and have been widely studied. Earlier results on integral representation of functionals on spaces of differentiable maps needed additional hypotheses to work, typically growth conditions on the derivatives.
scientific article; zbMATH DE number 492779