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In this paper we will review the main results concerning the issue ofstability for the determination unknown boundary portion of a thermicconducting body from Cauchy data for parabolic equations. We give detailed andselfcontained proofs. We prove that such problems are severely ill-posed in thesense that under a priori regularity assumptions on the unknown boundaries, upto any finite order of differentiability, the continuous dependence of unknownboundary from the measured data is, at best, of logarithmic type.

inverse problems

Quantitative estimates of unique continuation for parabolic equations,  determination of unknown time-varying boundaries and optimal stability  estimates