Integral representation of local functionals
The author answers to the following question: Under which conditions there exists a Borel function \(f(x,y)\), lower semicontinuous with respect to its second variable, such that a general functional \(F: X\times {\mathcal B}\to ]-\infty, +\infty]\) may be represented in the form \[ F(u,B)= \int_ B f(x, D^ k u(x))dx \] for every \(B\in {\mathcal B}\) and for every \(u\in X\), with \(k\) fixed positive integer. In order to prove the result he assumes an invariance condition of the type: \(F(u,B)= F(u+p, B)\) for all polynomials \(p: \mathbb{R}^ N\to \mathbb{R}^ M\) with \(\deg p<k\). The technique is very general and allows him to deal with functionals defined on \(L^ p\) or on every order Sobolev spaces as much. His result is an extension of known results obtained by Buttarro and Dal Maso.
scientific article; zbMATH DE number 590098
scientific article