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Determining coefficients in a class of heat equations via boundary measurements. (English) Zbl 0981.35096

Summary: When \(\Omega \subset{\mathbb R}^N\) is a bounded domain, we consider the problem of identifiability of the coefficients \(\rho,A,q\) in the equation \[ \rho(x)\partial_tu-\text{div}(A(x)\nabla u)+q(x)u=0 \] from boundary measurements on two pieces \(\Gamma_{\text{in}}\) and \(\Gamma_{\text{out}}\) of \(\partial \Omega\). Provided that \(\Gamma_{\text{in}} \cap \Gamma_{\text{out}}\) has a nonempty interior, and assuming that \(f(t,\sigma)\) is the given input datum for \((t,\sigma)\in(0,T)\times \Gamma_{\text{in}}\), and that the corresponding output datum is the thermal flux \(A(\sigma)\nabla u(T_0,\sigma)\cdot {\mathbf n}(\sigma)\), measured at a given time \(T_0\) for \(\sigma \in \Gamma_{\text{out}}\), we prove that knowledge of all possible pairs of input-output data \[ (f,A\nabla u(T_0)\cdot {\mathbf n}_{\mid \Gamma_{\text{out}}}) \] 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 results hold when a mean value of the thermal flux is measured over a small interval of time.

MSC:

35R30 Inverse problems for PDEs
35K15 Initial value problems for second-order parabolic equations
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