$\mathrm{\Gamma }$-limits of functionals determined by their infima

We study the integral representation of $\mathrm{\Gamma }$-limits of $p$-coercive integral functionals of the calculus of variations in the spirit of \cite{dalmaso-modica86}. We use infima of local Dirichlet problems to characterize the limit integrands. Applications to homogenization and relaxation are given.