A question, which arises frequently in shape optimal design, is
the convergence of domains. Usually the Hausdorff metric or
Hausdorff complementary metric is used in the potential topology
of QTR R^N. Unfortunately the convergence of domains does not
include convergence of its boundary. This fact can cause
difficulties if considering PDEs, defined on the domains and with
boundary conditions on its boundary. In this case a series of
solutions of the PDEs may not converge to the solution of the PDE
in the limit domain. Various restrictions on the set of domains,
like for example a cone property or Lipschitz boundaries have been
used in order to avoid difficulties of this kind. In this paper a
criterion for a set of domains is defined, such that from the
convergence of the domains follows convergence of the boundaries,
if one is restricting to this set of domains. Moreover it is
proved that this criterion is sharp and a similar criterion for
the convergence of the Lebesgue measure of the boundaries is given.

Number of pages | 20 |
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Publication status | Published - 1996 |
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