ncku-talk2010-11-22

NCTS(South)/ NCKU Math Colloquium

DATE 2010-11-22　16:10-17:00

PLACE R203, 2F, NCTS, NCKU

SPEAKER Gen Nakamura（math of hokudai）

TITLE Gradient estimate of solutions to elliptic/parabolic equations with discontinuous coefficients and its application

ABSTRACT 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 temper ture 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.

NCTS(South)/ NCKU Math Colloquium

DATE	2010-11-22　16:10-17:00

PLACE	R203, 2F, NCTS, NCKU

SPEAKER	Gen Nakamura（math of hokudai）

TITLE	Gradient estimate of solutions to elliptic/parabolic equations with discontinuous coefficients and its application

ABSTRACT	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 temper ture 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.