Determining Coefficients in a Class of Heat Equations via Boundary Measurements

When $\Omega \subset {\Bbb R}^N$ is a bounded domain, we consider the problem of identifiability of the coefficients $\rho ,A,q$ in the equation $\rho (x)\partial _tu-\mbox{div}(A(x)\nabla u)+q(x)u=0$ from boundary measurements on two pieces $\Gamma _{{\rm in}}$ and $\Gamma _{% {\rm out}}$ of $\partial \Omega $. Provided that $\Gamma _{{\rm in}}$ $\cap $ $\Gamma _{{\rm out}}$ has a nonempty interior, and assuming that $% f(t,\sigma )$ is the given input datum for $(t,\sigma )\in (0,T)\times \Gamma _{{\rm in}}$ and that the corresponding output datum is the thermal flux $A(\sigma )\nabla u(T_0,\sigma )\cdot \mbox{{\bf n}}(\sigma )$ measured at a given time $T_0$ for $\sigma \in \Gamma _{{\rm % out}}$, we prove that knowledge of all possible pairs of input-output data (f,A\nabla u(T_0)\cdot \mbox{{\bf n}}_{\mid \Gamma _{{\rm out}}}) determines uniquely the boundary spectral data of the underlying elliptic operator. Under suitable hypothesis on $\rho ,A,q$, their identifiability is then proved. The same resu...