In this post I would like to illustrate some ideas about solving TAT problem. Maybe this could be some breakthrough, I know it is tough.

In Justin’s paper, he followed his tutor’s idea, and showed that:

When \sup\sqrt{2\mu+\lambda}<3\inf\sqrt{\lambda}, then this problem could be solved under Isakov and  Tataru’s theorem’s hypothesis. Here \lambda,\mu are Lame coefficients.

And in his paper, he utilized two main tools.

1. Theorem 3.4.1 from Isakov’s book “Inverse Problem of Partial Differential Equations”.

2. Theorem 1 of Gunther Uhlmann’s paper “Thermoacoutic tomography with variable sound speed”.

However, it is a nice work. My question is:

1. How to relax the ratio \displaystyle\frac{\sup\sqrt{2\mu+\lambda}}{\inf\sqrt{\lambda}}, can it be very large?

2. If it is possible, can the ratio be infinity? Absolutely no, but, is there a possible way to give a bound or prove the boundness?

3. If still possible, how can we alter Isakov’s proof, as we can see, the Theorem 3.4.1  in the book is the result of Theorem 3.2.1′.

I have some ideas on modifications of the theorems. maybe we should begin with the definition of (strong) pseudo convexity.

[12.03.13  11:53 PM]After some thoughts, I have to say, if I follow Justin’s method with very rough estimate on the propagation, I will not have progress in this way. Because there is something I cannot avoid.



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