Quantitative estimates of unique continuation for parabolic equations and inverse initial-boundary value problems with unknown boundaries (Q2759066)

The authors are concerned with several questions related to the following initial and boundary value problems NEWLINE\[NEWLINE\begin{aligned} b(x)^jD_tu(x,t) - \text{div}(\kappa(x)\nabla u(x,t))=0,&\quad (x,t)\in \Omega \times (0,T],\;j=0,1,\\ u(x,0)=f(x), &\quad x\in \Omega,\\ \kappa(x)\nabla u(x,t)\cdot \nu(x)=g(x,t), &\quad (x,t)\in \partial \Omega \times (0,T], \end{aligned}NEWLINE\]NEWLINE where \(\Omega\subset {\mathbb R}^n\) is a (smooth) bounded domain and \(\nu(x)\) denotes the outer unit normal at \(x\in \partial \Omega\). Moreover, the scalar function \(b\) and the thermal conductivity \(n\times n\) matrix \(\kappa\) are assumed to be Lipschitz continuous and, respectively, strictly positive and uniformly positive definite on \({\overline \Omega}\). NEWLINENEWLINENEWLINEThe paper is rich in many interesting results and gives a complete description of topics concerning the state-of-the-art topics, such as the so-called three spheres and three cylinders inequalities (both in the interior and on the boundary of \(\Omega\)), such as the strong uniqueness continuation on the characteristic planes \(t=t_0\), from Cauchy data as well as from time-like surfaces. A basic technique used by the authors is the so-called elliptic continuation of solutions to parabolic equations. NEWLINENEWLINENEWLINEThe last topic dealt with is concerned with recovering some inaccessible part \(I\subset \partial \Omega\) from additional measurements of the temperature \(u\) on an accessible part \(A\subset \partial \Omega\) under the assumption that the heat flux \(g\) vanishes on \(I\). The main results are concerned with the stability of the map ``data \(\to {\overline \Omega}\)''. Under suitable assumptions the authors show that the continuity modulus is, as in the elliptic case, of logarithmic type.