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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

Gianluca Mola

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

K. -C. Chang, Methods in Nonlinear Analysis, Monographs in Mathematics, Springer-Verlag, Berlin, 2005.  Google Scholar

[6]

D. Gale and H. Nikaido, The jacobian matrix and global univalence of mappings, Math. Annalen, 159 (1965), 81-93.  doi: 10.1007/BF01360282.  Google Scholar

[7]

T. Kato, Perturbation Theory for Linear Operators, Grundlehren der mathematischen Wissenschaften, 132, Springer-Verlag, Berlin and New York, 1966; 2nd ed., 1976.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

L. Lorenzi, An identification problem for the Ornstein-Uhlenbeck operator, Journal of Inverse and Ill-posed Problems, 19 (2011), 293-326.   Google Scholar

[12]

A. Sh. Lyubanova, Identification of a constant coefficient in an elliptic equation, Appl. Anal., 87 (2008), 1121-1128.  doi: 10.1080/00036810802189654.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

K. -C. Chang, Methods in Nonlinear Analysis, Monographs in Mathematics, Springer-Verlag, Berlin, 2005.  Google Scholar

[6]

D. Gale and H. Nikaido, The jacobian matrix and global univalence of mappings, Math. Annalen, 159 (1965), 81-93.  doi: 10.1007/BF01360282.  Google Scholar

[7]

T. Kato, Perturbation Theory for Linear Operators, Grundlehren der mathematischen Wissenschaften, 132, Springer-Verlag, Berlin and New York, 1966; 2nd ed., 1976.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

L. Lorenzi, An identification problem for the Ornstein-Uhlenbeck operator, Journal of Inverse and Ill-posed Problems, 19 (2011), 293-326.   Google Scholar

[12]

A. Sh. Lyubanova, Identification of a constant coefficient in an elliptic equation, Appl. Anal., 87 (2008), 1121-1128.  doi: 10.1080/00036810802189654.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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