Relaxation of Non-Convex Integral Functionals in the Multidimensional Scalar Case

We study integral functionals defined on scalar Sobolev spaces of the form $$E[f]:u\mapsto {\int }_{\mathrm{\Omega }}f(x,u(x),\mathrm{\nabla }u(x))dx,$$ 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.