Cloaking via change of variables for the Helmholtz equation

Job Candidate Talk
Wednesday, January 12, 2011 - 2:00pm for 1 hour (actually 50 minutes)
Skiles 006
Nguyen Hoai-Minh – Courant Institute of Mathematical Sciences
Wilfrid Gangbo
A region of space is cloaked for a class of measurements if observers are not only unaware of its contents, but also unaware of the presence of the cloak using such measurements. One approach to cloaking is the change of variables scheme introduced by Greenleaf, Lassas, and Uhlmann for electrical impedance tomography and by Pendry, Schurig, and Smith for the Maxwell equations. They used a singular change of variables which blows up a point into the cloaked region. To avoid this singularity, various regularized schemes have been proposed. In this talk I present results related to cloaking via change of variables for the Helmholtz equation using the natural regularized scheme introduced by Kohn, Shen, Vogelius, and Weintein, where the authors used a transformation which blows up a small ball instead of a point into the cloaked region. I will discuss the degree of invisibility for a finite range or the full range of frequencies, and the possibility of achieving perfect cloaking. At the end of my talk, I will also discuss some results related to the wave equation in 3d.