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

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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,$

and the additional conditions

$\left\langle A_{1} u(T),u(T)\right\rangle = \varphi_{1}, ~~~··· ~~~,\left\langle A_{n} u(T),u(T)\right\rangle = \varphi_{n},$

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.

Keywords: Identification problems, unknown constants, linear evolution equations in Hilbert spaces, linear parabolic equations, well-posedness results.

Mathematics Subject Classification: Primary: 35R30, 35K90; Secondary: 35K20, 35K25, 65N30.

Citation: Gianluca Mola. Recovering a large number of diffusion constants in a parabolic equation from energy measurements. Inverse Problems and Imaging, 2018, 12 (3) : 527-543. doi: 10.3934/ipi.2018023

References:

[1]	M. Akamatsu, G. Nakamura and S. Steinberg, Identification of Lam6 coefficients from boundary observations, Inverse Problems, 7 (1991), 335-354. doi: 10.1088/0266-5611/7/3/003.
[2]	E. A. Artyukhin and A. S. Okhapkin, Determination of the parameters in the generalized heat-conduction equation from transient experimental data, J. Eng. Phys. Thermophys., 42 (1982), 693-698. doi: 10.1007/BF00835106.
[3]	H. Brezis, Analyse Fonctionnelle, Théorie et Applications, Masson, Paris, 1983; (English Translation) Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011.
[4]	J. R. Cannon, Determination of certain parameters in heat conduction problems, J. Math. Anal. Appl., 8 (1964), 188-201. doi: 10.1016/0022-247X(64)90061-7.
[5]	K. -C. Chang, Methods in Nonlinear Analysis, Monographs in Mathematics, Springer-Verlag, Berlin, 2005.
[6]	D. Gale and H. Nikaido, The jacobian matrix and global univalence of mappings, Math. Annalen, 159 (1965), 81-93. doi: 10.1007/BF01360282.
[7]	T. Kato, Perturbation Theory for Linear Operators, Grundlehren der mathematischen Wissenschaften, 132, Springer-Verlag, Berlin and New York, 1966; 2nd ed., 1976.
[8]	A. Lorenzi, Recovering two constants in a parabolic linear equation, Journal of Physics: Conference Series, 73 (2007). doi: 10.1088/1742-6596/73/1/012014.
[9]	A. Lorenzi and G. Mola, Identification of a real constant in linear evolution equations in Hilbert spaces, Inverse Problems and Imaging, 5 (2011), 695-714. doi: 10.3934/ipi.2011.5.695.
[10]	A. Lorenzi and G. Mola, Recovering the reaction and the diffusion coefficients in a linear parabolic equation, Inverse Problems, 28 (2012), 075006, 23 pp.
[11]	L. Lorenzi, An identification problem for the Ornstein-Uhlenbeck operator, Journal of Inverse and Ill-posed Problems, 19 (2011), 293-326.
[12]	A. Sh. Lyubanova, Identification of a constant coefficient in an elliptic equation, Appl. Anal., 87 (2008), 1121-1128. doi: 10.1080/00036810802189654.
[13]	G. Mola, Identification of the diffusion coefficient in linear evolution equations in Hilbert spaces, J. Abstract Differential Equations and Applications, 2 (2011), 14-28.
[14]	G. Mola, N. Okazawa and T. Yokota, Reconstruction of two constant coefficients in linear anisotropic diffusion model, Inverse Problems, 32 (2016), 115016, 22 pp.
[15]	G. Mola, N. Okazawa, J. Prüss and T. Yokota, Semigroup-theoretic approach to identification of linear diffusion coefficients, Discrete Continuous Dynamical Systems, Series S, 9 (2016), 777-790. doi: 10.3934/dcdss.2016028.
[16]	G. Nakamura and G. Uhlmann, Identification of Lame parameters by boundary measurements, American Journal of Mathematics, 115 (1993), 1161-1187. doi: 10.2307/2375069.
[17]	N. Okazawa, On the perturbation of linear operators in Banach and Hilbert spaces, J. Math. Soc. Japan, 34 (1982), 677-701. doi: 10.2969/jmsj/03440677.
[18]	S. J. L. van Eijndhoven and J. de Graaf, A Fundamental Approach to the Generalized Eigenvalue Problem for Self-Adjoint Operators, J. Functional Analysis, 63 (1985), 74-85. doi: 10.1016/0022-1236(85)90098-9.
[19]	M. Yamamoto, Determination of constant parameters in some semilinear parabolic equations, Ill-posed Problems in Natural Sciences (Moscow, 1991), VSP, Utrecht, 1992,439-445.

show all references

References:

[1]	M. Akamatsu, G. Nakamura and S. Steinberg, Identification of Lam6 coefficients from boundary observations, Inverse Problems, 7 (1991), 335-354. doi: 10.1088/0266-5611/7/3/003.
[2]	E. A. Artyukhin and A. S. Okhapkin, Determination of the parameters in the generalized heat-conduction equation from transient experimental data, J. Eng. Phys. Thermophys., 42 (1982), 693-698. doi: 10.1007/BF00835106.
[3]	H. Brezis, Analyse Fonctionnelle, Théorie et Applications, Masson, Paris, 1983; (English Translation) Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011.
[4]	J. R. Cannon, Determination of certain parameters in heat conduction problems, J. Math. Anal. Appl., 8 (1964), 188-201. doi: 10.1016/0022-247X(64)90061-7.
[5]	K. -C. Chang, Methods in Nonlinear Analysis, Monographs in Mathematics, Springer-Verlag, Berlin, 2005.
[6]	D. Gale and H. Nikaido, The jacobian matrix and global univalence of mappings, Math. Annalen, 159 (1965), 81-93. doi: 10.1007/BF01360282.
[7]	T. Kato, Perturbation Theory for Linear Operators, Grundlehren der mathematischen Wissenschaften, 132, Springer-Verlag, Berlin and New York, 1966; 2nd ed., 1976.
[8]	A. Lorenzi, Recovering two constants in a parabolic linear equation, Journal of Physics: Conference Series, 73 (2007). doi: 10.1088/1742-6596/73/1/012014.
[9]	A. Lorenzi and G. Mola, Identification of a real constant in linear evolution equations in Hilbert spaces, Inverse Problems and Imaging, 5 (2011), 695-714. doi: 10.3934/ipi.2011.5.695.
[10]	A. Lorenzi and G. Mola, Recovering the reaction and the diffusion coefficients in a linear parabolic equation, Inverse Problems, 28 (2012), 075006, 23 pp.
[11]	L. Lorenzi, An identification problem for the Ornstein-Uhlenbeck operator, Journal of Inverse and Ill-posed Problems, 19 (2011), 293-326.
[12]	A. Sh. Lyubanova, Identification of a constant coefficient in an elliptic equation, Appl. Anal., 87 (2008), 1121-1128. doi: 10.1080/00036810802189654.
[13]	G. Mola, Identification of the diffusion coefficient in linear evolution equations in Hilbert spaces, J. Abstract Differential Equations and Applications, 2 (2011), 14-28.
[14]	G. Mola, N. Okazawa and T. Yokota, Reconstruction of two constant coefficients in linear anisotropic diffusion model, Inverse Problems, 32 (2016), 115016, 22 pp.
[15]	G. Mola, N. Okazawa, J. Prüss and T. Yokota, Semigroup-theoretic approach to identification of linear diffusion coefficients, Discrete Continuous Dynamical Systems, Series S, 9 (2016), 777-790. doi: 10.3934/dcdss.2016028.
[16]	G. Nakamura and G. Uhlmann, Identification of Lame parameters by boundary measurements, American Journal of Mathematics, 115 (1993), 1161-1187. doi: 10.2307/2375069.
[17]	N. Okazawa, On the perturbation of linear operators in Banach and Hilbert spaces, J. Math. Soc. Japan, 34 (1982), 677-701. doi: 10.2969/jmsj/03440677.
[18]	S. J. L. van Eijndhoven and J. de Graaf, A Fundamental Approach to the Generalized Eigenvalue Problem for Self-Adjoint Operators, J. Functional Analysis, 63 (1985), 74-85. doi: 10.1016/0022-1236(85)90098-9.
[19]	M. Yamamoto, Determination of constant parameters in some semilinear parabolic equations, Ill-posed Problems in Natural Sciences (Moscow, 1991), VSP, Utrecht, 1992,439-445.

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