Asymptotic expansion of the solution of an interface problem in a polygonal domain with thin layer

We consider the solution of an interface problem posed in a bounded domain coated with a layer of thickness ε and with external boundary conditions of Dirichlet or Neumann type. Our aim is to build a multi-scale expansion as ε goes to 0 for that solution. After presenting a complete multi-scale expansion in a smooth coated domain, we focus on the case of a corner domain. Singularities appear, obstructing the construction of the expansion terms in the same way as in the smooth case. In order to take these singularities into account, we construct profiles in an infinite coated sectorial domain. Combining expansions in the smooth case with splittings in regular and singular parts involving the profiles, we construct two families of multi-scale expansions for the solution in the coated domain with corner. We prove optimal estimates for the remainders of the multiscale expansions.