Lipschitz stability in the determination of the principal part ofa parabolic equation

Abstract

Let y(h)(t,x) be one solution to \[\partial_t y(t,x) - \sum_{i, j=1}^{n}\partial_{j} (a_{ij}(x)\partial_i y(t,x))= h(t,x), \thinspace 0<t<T, \thinspace x\in \Omega\] with a non-homogeneous term h, and $y\vert_{(0,T)\times\partial\Omega} = 0$ ,where $\Omega \subset\Bbb R^n$ is a bounded domain. We discuss an inverse problemof determining n(n+1)/2 unknown functions a _ij by $\{ \partial_{\nu}y(h_{\ell})\vert_{(0,T)\times \Gamma_0}$ , $y(h_{\ell})(\theta,\cdot)\}_{1\le \ell\le \ell_0}$ after selecting input sources $h_1, ...,h_{\ell_0}$ suitably, where $\Gamma_0$ is an arbitrary subboundary, $\partial_{\nu}$ denotes the normal derivative, $0 < \theta < T$ and $\ell_0 \in \Bbb N$ . In the case of $\ell_0 = (n+1)^2n/2$ , we provethe Lipschitz stability in the inverse problem if we choose $(h_1, ...,h_{\ell_0})$ from a set ${\cal H} \subset \{ C_0^{\infty}((0,T)\times \omega)\}^{\ell_0}$ with an arbitrarily fixed subdomain $\omega \subset \Omega$ . Moreover we can take $\ell_0 = (n+3)n/2$ by making special choices for $h_{\ell}$ , $1 \le \ell \le \ell_0$ . The proof is based on a Carlemanestimate.