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Asymptotic expansion of the solution of an interface problem in a polygonal domain with thin layer. (English) Zbl 1136.35021
The authors consider the solution of an interface problem posed in a bounded domain coated with a layer of thickness \(\varepsilon\) and subjected to external boundary conditions of Dirichlet or Neumann type. The main goal is to build a multi-scale expansion for solutionof the problem as \(\varepsilon\) goes to \(0\). After presenting a complete multi-scale expansion in a smooth coated domain, the authors consider the case of a corner domain. Appeared singularities obstruct the derivation of the expansion terms in the same way as in the smooth case. In order to take these singularities into account, the authors construct profiles in an infinite coated sectorial domain. Combining expansions in the smooth case with splittings in regular and singular parts involving the profiles, the authors obtain two families of multi-scale expansions for the solution in the coated domain with corner. Optimal estimates are proved for the remainders of the multi-scale expansions.

35J25 Boundary value problems for second-order elliptic equations
35C20 Asymptotic expansions of solutions to PDEs