Seminar

An interior gradient estimate of solutions of parabolic
operators with discontinuous coefficients
and its application to gradient estimates of the fundamental solutions
of these operators are given in my talk. The discontinuities of
coefficients are across several closed surfaces and some of them can
touch each other. The interior estimate is the parabolic version
of Li-Vogelius and Li-Nirenberg results. Since the parabolic operators
with discontinuous coefficients can model the temperature distribution
in heat conductors with inclusions, the result could be useful also
for inverse problem such as identifying unknown inclusions which can
touch each other. More precisely, what I have in mind is to extend the
so-called dynamical probe method which is known as a method to
reconstruct the unknown discontinuities of the media, such as cavities
and inclusions, to the case that some of the inclusions can touch each
other. I will explain how to extract the dominant part of the
reflected solution by using the gradient estimate of the fundamental
solution. But in order to establish the dynamical probe method for
this case, I still need to prove the estimate from below of the
modulus of the dominant part of reflected solution.