Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-25T08:22:59.688Z Has data issue: false hasContentIssue false
  • English
  • Français

Exponential asymptotic stability for an elliptic equation with memory arising in electrical conduction in biological tissues

Published online by Cambridge University Press:  20 August 2009

M. AMAR ,
D. ANDREUCCI ,
P. BISEGNA  and
R. GIANNI
M. AMAR
Affiliation:
Dipartimento di Metodi e Modelli Matematici, Università di Roma ‘La Sapienza’, via A. Scarpa 16, 00161 Roma, Italy email: [email protected], [email protected], [email protected]
D. ANDREUCCI
Affiliation:
Dipartimento di Metodi e Modelli Matematici, Università di Roma ‘La Sapienza’, via A. Scarpa 16, 00161 Roma, Italy email: [email protected], [email protected], [email protected]
P. BISEGNA
Affiliation:
Dipartimento di Ingegneria Civile, Università di Roma ‘Tor Vergata’, via del Politecnico 1, 00133 Roma, Italy email: [email protected]
R. GIANNI
Affiliation:
Dipartimento di Metodi e Modelli Matematici, Università di Roma ‘La Sapienza’, via A. Scarpa 16, 00161 Roma, Italy email: [email protected], [email protected], [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We study an electrical conduction problem in biological tissues in the radiofrequency range, which is governed by an elliptic equation with memory. We prove the time exponential asymptotic stability of the solution. We provide in this way both a theoretical justification to the complex elliptic problem currently used in electrical impedance tomography and additional information on the structure of the complex coefficients appearing in the elliptic equation. Our approach relies on the fact that the elliptic equation with memory is the homogenisation limit of a sequence of problems for which we prove suitable uniform estimates.

Type
Papers
Information
European Journal of Applied Mathematics , Volume 20 , Issue 5 , October 2009 , pp. 431 - 459
Copyright
Copyright © Cambridge University Press 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Amar, M., Andreucci, D., Bisegna, P. & Gianni, R. (2004) An elliptic equation with history. C. R. Acad. Sci. Paris, Ser. I 338, 595–598.CrossRefGoogle Scholar
[2]Amar, M., Andreucci, D., Bisegna, P. & Gianni, R. (2004) Evolution and memory effects in the homogenization limit for electrical conduction in biological tissues. Math. Models Methods Appl. Sci. 14, 1261–1295.CrossRefGoogle Scholar
[3]Amar, M., Andreucci, D., Bisegna, P. & Gianni, R. (2005) Existence and uniqueness for an elliptic problem with evolution arising in electrodynamics. Nonlin. Anal. Real World Appl. 6, 367–380.CrossRefGoogle Scholar
[4]Amar, M., Andreucci, D., Bisegna, P. & Gianni, R. (2006) On a hierarchy of models for electrical conduction in biological tissues. Math. Methods Appl. Sci. 29, 767–787.CrossRefGoogle Scholar
[5]Ambrosio, L., Fusco, N. & Pallara, D. (2000) Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs, Oxford University Press, Oxford.CrossRefGoogle Scholar
[6]Appleby, J. A. D., Fabrizio, M., Lazzari, B. & Reynolds, D. W. (2006) On exponential asymptotic stability in linear viscoelasticity. Math. Models Methods Appl. Sci. 16, 1677–1694.CrossRefGoogle Scholar
[7]Bayford, R. H. (2006) Bioimpedance tomography (electrical impedance tomography). Annu. Rev. Biomed. Eng. 8, 63–91.CrossRefGoogle ScholarPubMed
[8]Borcea, L. (2002) Electrical impedance tomography. Inverse Prob. 18, R99–R136.CrossRefGoogle Scholar
[9]Cioranescu, D. & Saint Jean Paulin, J. (1999) Homogenization of Reticulated Structures, Applied Mathematical Sciences, Vol. 136. Springer-Verlag, New York.CrossRefGoogle Scholar
[10]Dehghani, H. & Soni, N. K. (2005) Electrical impedance spectroscopy: Theory. In Paulsen, K. D., Meaney, P. M. & Gilman, L. C. (editors), Alternative Breast Imaging: Four Model-Based Approaches, Springer, pp. 85–105.CrossRefGoogle Scholar
[11]Fabrizio, M. (1985) Sul problema di Fichera in viscoelasticità lineare. In: Atti del III Meeting: Waves and Stability, 7–12 October, 1985, Bari, Italy.Google Scholar
[12]Fabrizio, M. (1992) Sulla correttezza di un problema integrodifferenziale della viscoelasticità. In: Seminari di Analisi, 1992, Dipartimento di Matematica, Università di Bologna, Bologna, Italy.Google Scholar
[13]Fabrizio, M. & Lazzari, B. (1990) Sulla stabilità di un sistema viscoelastico lineare. In: Acc. Naz. Lincei, Tavola rotonda sul tema: Continui con Memoria, 1990, Roma, Italy.Google Scholar
[14]Fabrizio, M. & Morro, A. (1998) Viscoelastic relaxation functions compatible with thermodynamics. J. Elast. 19, 63–75.CrossRefGoogle Scholar
[15]Fichera, G. (1979) Avere una memoria tenace crea gravi problemi. Arch. Ration. Mech. Anal. 70, 101–112.CrossRefGoogle Scholar
[16]Giorgi, C., Naso, M. G. & Pata, V. (2001) Exponential stability in linear heat conduction with memory: A semigroup approach. Comm. Appl. Anal. 5, 121–133.Google Scholar
[17]Hummel, H.-K. (2000) Homogenization for heat transfer in polycrystals with interfacial resistances. Appl. Anal. 75, 403–424.CrossRefGoogle Scholar
[18]Lazzari, B. & Nibbi, R. (1992) Sufficient conditions for the exponential stability in linear conductors with memory. Int. J. Eng. Sci. 30, 533–544.CrossRefGoogle Scholar
[19]Lene, F. & Leguillon, D. (1981) Étude de l'influence d'un glissement entre les constituants d'un matériau composite sur ses coefficients de comportement effectifs. J Méc. 20, 509–536.Google Scholar
[20]Medjden, M. & Tatar, N.-E. (2005) Asymptotic behavior for a viscoelastic problem with not necessarily decreasing kernel. Appl. Math. Comput. 167, 1221–1235.Google Scholar
[21]Raviart, P. A. & Thomas, J.-M. (1983) Introduction á l'analyse numérique des équations aux dérivées partielles, Masson, Paris.Google Scholar
[22]Slemrod, M. (1981) Global existence, uniqueness, and asymptotic stability of classical smooth solutions in one-dimensional nonlinear thermoelasticity. Arch. Ration. Mech. Anal. 76, 97–133.CrossRefGoogle Scholar
[23]Taylor, A. E. & Lay, D. C. (1980) Introduction to Functional Analysis, 2nd ed., John Wiley, New York, Chichester and Brisbane.Google Scholar