### A singularly perturbed mixed boundary value problem

*Martin Costabel and Monique Dauge*

We study a mixed Neumann-Robin boundary value problem for the Laplace
operator in a smooth domain in **R**^2. The Robin condition contains a
parameter ** ε **
and tends to a Dirichlet condition as
** ε ** --> 0.
We give a complete asymptotic expansion of the
solution in powers of ** ε **.
At the points where the boundary
conditions change, there appear boundary layers of corner type of size
** ε **. They describe how the singularities of the limit
Dirichlet-Neumann problem are approximated. We give sharp estimates in
various Sobolev norms and show in particular that there exist terms of
order O(** ε **
log** ε **).

Published in *Comm. Partial Differential Equations* **21** (11-12), 1996, 1919--1949.