  The time domain enclosure method is one of analytical methods for inverse
obstacle problems governed by partial differential equations in the time
domain. This paper considers the case when the governing equation is given by
the Maxwell system and consists of two parts. The first part establishes the
base of the time domain enclosure method for the Maxwell system using a single
set of the solutions over a finite time interval for a general (isotropic)
inhomogeneous medium in the whole space. It is a system of asymptotic
inequalities for the indicator function which may enable us to apply the time
domain enclosure method to the problem of finding unknown penetrable obstacles
embedded in various background media. As a first step of its expected
applications, the case when the background medium is homogeneous and isotropic,
is considered and the time domain enclosure method is realized. This is the
second part.

Analysis of PDEs