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Generalized Impedance Boundary Conditions for Strongly Absorbing Obstacles: the full Wave Equations
This paper is devoted to the study of the generalized impedance boundary
conditions (GIBCs) for a strongly absorbing obstacle in the {\bf time} regime
in two and three dimensions. The GIBCs in the time domain are heuristically
derived from the corresponding conditions in the time harmonic regime. The
latters are frequency dependent except the one of order 0; hence the formers
are non-local in time in general. The error estimates in the time regime can be
derived from the ones in the time harmonic regime when the frequency dependence
is well-controlled. This idea is originally due to Nguyen and Vogelius in
\cite{NguyenVogelius2} for the cloaking context. In this paper, we present the
analysis to the GIBCs of orders 0 and 1. To implement the ideas in
\cite{NguyenVogelius2}, we revise and extend the work of Haddar, Joly, and
Nguyen in \cite{HJNg1}, where the GIBCs were investigated for a fixed frequency
in three dimensions. Even though we heavily follow the strategy in
\cite{NguyenVogelius2}, our analysis on the stability contains new ingredients
and ideas. First, instead of considering the difference between solutions of
the exact model and the approximate model, we consider the difference between
their derivatives in time. This simple idea helps us to avoid the machinery
used in \cite{NguyenVogelius2} concerning the integrability with respect to
frequency in the low frequency regime. Second, in the high frequency regime,
the Morawetz multiplier technique used in \cite{NguyenVogelius2} does not fit
directly in our setting. Our proof makes use of a result by H\"ormander in
\cite{Hor}. Another important part of the analysis in this paper is the
well-posedness in the time domain for the approximate problems imposed with
GIBCs on the boundary of the obstacle, which are non-local in time
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