| Abstract: |
| Let $\Omega$ be a bounded domain of $\mathbb{R}^n$ with smooth boundary $\partial \Omega$. We study the inverse problem consisting in the identification of the function $u:(0,\infty) \times\Omega \to \mathbb{R}$ and the $n \times n$ symmetric and positive definite matrix $\mathbb{A}$
({\it diffusion matrix})
that fulfill the parabolic problem
$$
\begin{cases}
\dfrac{\partial}{\partial t} u - \nabla \cdot \mathbb{A} \nabla u = 0 \quad \text{in} \quad (0,\infty) \times \Omega,\
u(0,\cdot) = \phi \quad \text{in} \quad \Omega,\
u(t,\cdot) = 0 \quad \text{on} \quad (0,\infty) \times \partial\Omega,
\end{cases}
$$
along with the additional integral measurements at a fixed time $T > 0$
$$
\int_\Omega\dfrac{\partial}{\partial x_i}u(T,\boldsymbol{x}) \cdot \dfrac{\partial}{\partial x_j}u(T,\boldsymbol{x}) d \boldsymbol{x}
=
\mathbb{M}_{i,j},
\quad 1 \leq i \leq j \leq n.
$$
Under suitable assumptions on the initial datum $\phi:\Omega \to \mathbb{R}$ and on the overdeterminating conditions $\mathbb{M}_{i,j} \in \mathbb{R}$, we shall prove that the solution $(u,\mathbb{A})$ of such a problem is unique and depends continuously (Lipschitz) on the data $(\phi,\mathbb{M}_{i,j})$. |
|