A new approach to Young measure theory, relaxation and convergence in energy
The paper contains a detailed presentation of the theory of Young measures, with particular emphasis on their application to the study of lower semicontinuity of integral functionals of the calculus of variations of the form \[ I(u)= \int_\Omega L(x,u,\nabla u) dx.\tag{1} \] Section 2 deals with some properties of functions in Sobolev spaces. Section 3 is devoted to Young measures and to the characterization of them as measurable functions into a suitable metric space. In Section 4 the theory of gradient Young measures is presented together with a survey on the already existing literature. Section 5 finally deals with some applications of Young measures to the study of the asymptotic behaviour of integral functionals of the form (1) on weakly convergent sequences of Sobolev functions, which allows to obtain relaxation results under minimal assumptions on the integrand \(L\).
scientific article; zbMATH DE number 1394706
scientific article