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

TL;DR

This paper reviews stability issues in determining unknown boundaries of a thermic body from parabolic data, showing the problem is severely ill-posed with logarithmic stability limits.

Contribution

It provides detailed proofs that the boundary determination problem is severely ill-posed with logarithmic stability, under regularity assumptions.

Findings

Problems are severely ill-posed with logarithmic stability

Stability estimates depend on boundary regularity

Provides detailed, self-contained proofs

Abstract

In this paper we will review the main results concerning the issue of stability for the determination unknown boundary portion of a thermic conducting body from Cauchy data for parabolic equations. We give detailed and selfcontained 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 unknown boundary from the measured data is, at best, of logarithmic type.