# American Institute of Mathematical Sciences

June  2018, 12(3): 527-543. doi: 10.3934/ipi.2018023

## Recovering a large number of diffusion constants in a parabolic equation from energy measurements

 Politecnico di Milano, Piazza Leonardo da Vinci 32, 2013 Milano, Italy

Received  April 2017 Revised  November 2017 Published  March 2018

Let
 $\left(H, \left\langle { \cdot , \cdot } \right\rangle \right)$
be a separable Hilbert space and
 $A_{i}:D(A_i) \to H$
(
 $i = 1,···,n$
) be a family of nonnegative and self-adjoint operators mutually commuting. We study the inverse problem consisting in the identification of a function
 $u:[0,T] \to H$
and
 $n$
constants
 $α_{1},···,α_{n} > 0$
(diffusion coefficients) that fulfill the initial-value problem
 $u'(t) + α_{1} A_{1}u(t) + ··· + α_{n} A_{n}u(t) = 0, ~~~t ∈ (0,T), ~~~u(0) = x,$
 $\left\langle A_{1} u(T),u(T)\right\rangle = \varphi_{1}, ~~~··· ~~~,\left\langle A_{n} u(T),u(T)\right\rangle = \varphi_{n},$
where
 $\varphi_{i}$
are given positive constants. Under suitable assumptions on the operators
 $A_{i}$
and on the initial data
 $x ∈ H$
, we shall prove that the solution of such a problem is unique and depends continuously on the data. We apply the abstract result to the identification of diffusion constants in a heat equation and of the Lamé parameters in a elasticity problem on a plate.
Citation: Gianluca Mola. Recovering a large number of diffusion constants in a parabolic equation from energy measurements. Inverse Problems & Imaging, 2018, 12 (3) : 527-543. doi: 10.3934/ipi.2018023
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