Relaxation of Non-Convex Integral Functionals in the Multidimensional Scalar Case
We study integral functionals defined on scalar Sobolev spaces of the form $$E[f]:u\mapsto \int_Ωf(x,u(x),\nabla u(x)) d x,$$ with an emphasis on the non-convex case, and the difficulties it involves to prevent the Lavrentiev phenomenon. We determine a formulation of the lower semicontinuous envelope of $E[f]$ with respect to various topologies and with fixed Lipschitz Dirichlet boundary conditions.
math.AP↗