  The H^2-regularity of variational solutions to a two-dimensional transmission
problem with geometric constraint is investigated, in particular when part of
the interface becomes part of the outer boundary of the domain due to the
saturation of the geometric constraint. In such a situation, the domain
includes some non-Lipschitz subdomains with cusp points, but it is shown that
this feature does not lead to a regularity breakdown. Moreover, continuous
dependence of the solutions with respect to the domain is established.
