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

In this article, we review the main results concerning the issue of stability for the determination of unknown boundary portions of a thermic conducting body from Cauchy data for parabolic equations. We give detailed and self-contained proofs. We prove that such problems are severely ill-posed in the sense that under a priori regularity assumptions on the unknown boundaries, up to any finite order of differentiability, the continuous dependence of an unknown boundary from the measured data is, at best, of logarithmic type. We review the main results concerning quantitative estimates of unique continuation for solutions to second-order parabolic equations. We give a detailed proof of a Carleman estimate crucial for the derivation of the stability estimates.