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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: amar@dmmm.uniroma1.it, andreucci@dmmm.uniroma1.it, gianni@dmmm.uniroma1.it
D. ANDREUCCI
Affiliation:
Dipartimento di Metodi e Modelli Matematici, Università di Roma ‘La Sapienza’, via A. Scarpa 16, 00161 Roma, Italy email: amar@dmmm.uniroma1.it, andreucci@dmmm.uniroma1.it, gianni@dmmm.uniroma1.it
P. BISEGNA
Affiliation:
Dipartimento di Ingegneria Civile, Università di Roma ‘Tor Vergata’, via del Politecnico 1, 00133 Roma, Italy email: bisegna@uniroma2.it
R. GIANNI
Affiliation:
Dipartimento di Metodi e Modelli Matematici, Università di Roma ‘La Sapienza’, via A. Scarpa 16, 00161 Roma, Italy email: amar@dmmm.uniroma1.it, andreucci@dmmm.uniroma1.it, gianni@dmmm.uniroma1.it
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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

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