Local and global uniform convergence for elliptic problems on varying domains

The aim of the paper is to prove optimal results on local and global uniform convergence of solutions to elliptic equations with Dirichlet boundary conditions on varying domains. We assume that the limit domain be stable in the sense of Keldyš [Amer. Math. Soc. Transl. 51 (1966) 1–73]. We further assume that the approaching domains satisfy a necessary condition in the inside of the limit domain, and only require L2-convergence outside. As a consequence, uniform and L2-convergence are the same in the trivial case of homogenisation of a perforated domain. We are also able to deal with certain cracking domains.