Comptes Rendus
no. 8
Partial Differential Equations
An elliptic equation with history
[Une equation elliptique avec histoire]

Présenté par : Philippe G. Ciarlet

Micol Amar1 ; Daniele Andreucci1 ; Paolo Bisegna2 ; Roberto Gianni1
1 Università di Roma“La Sapienza”, Dipartimento di Metodi e Modelli Matematici, Via A. Scarpa 16, 00161 Roma, Italy
2 Università di Roma “Tor Vergata”, Dipartimento di Ingegneria Civile, Via del Politecnico 1, 00133 Roma, Italy
Comptes Rendus. Mathématique, Volume 338 (2004) no. 8, pp. 595-598.

We prove the existence and uniqueness for a semilinear elliptic problem with memory, both in the weak and the classical setting. This problem describes the effective behaviour of a biological tissue under the injection of an electrical current in the radiofrequency range.

On démontre l'existence et l'unicité pour un problème elliptique semilinéaire avec mémoire, dans l'arrangement faible et classique. Ce problème décrit le comportement effective d'un tissu biologique sous l'injection d'un courant électrique dans le domaine des radiofréquences.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2004.02.008

Micol Amar 1 ; Daniele Andreucci 1 ; Paolo Bisegna 2 ; Roberto Gianni 1

1 Università di Roma“La Sapienza”, Dipartimento di Metodi e Modelli Matematici, Via A. Scarpa 16, 00161 Roma, Italy
2 Università di Roma “Tor Vergata”, Dipartimento di Ingegneria Civile, Via del Politecnico 1, 00133 Roma, Italy 
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Micol Amar; Daniele Andreucci; Paolo Bisegna; Roberto Gianni. An elliptic equation with history. Comptes Rendus. Mathématique, Volume 338 (2004) no. 8, pp. 595-598. doi : 10.1016/j.crma.2004.02.008. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.02.008/

[1] M. Amar; D. Andreucci; P. Bisegna; R. Gianni Homogenization limit for electrical conduction in biological tissues in the radio-frequency range, C. R. Mecanique, Volume 331 (2003), pp. 503-508

[2] M. Amar; D. Andreucci; P. Bisegna; R. Gianni Evolution and memory effects in the homogenization limit for electrical conduction in biological tissues, Math. Models Methods Appl. Sci., Volume 9 (2004) no. 14 (in press)

[3] M. Fabrizio An existence and uniqueness theorem in quasi-static viscoelasticity, Quart. Appl. Math., Volume 47 (1989), pp. 1-9

[4] G. Fichera Avere una memoria tenace crea gravi problemi, Arch. Rational Mech. Anal., Volume 70 (1972), pp. 101-112

[5] G. Fichera Sul principio di memoria evanescente, Rend. Sem. Mat. Univ. Padova, Volume 68 (1982), pp. 245-259

[6] D. Gilbarg; N.S. Trudinger Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983

  • M. Amar; D. Andreucci; R. Gianni; C. Timofte Concentration and homogenization in electrical conduction in heterogeneous media involving the Laplace–Beltrami operator, Calculus of Variations and Partial Differential Equations, Volume 59 (2020) no. 3 | DOI:10.1007/s00526-020-01749-x
  • Micol Amar; Daniele Andreucci; Roberto Gianni; Claudia Timofte A degenerate pseudo-parabolic equation with memory, Communications in Applied and Industrial Mathematics, Volume 10 (2019) no. 1, p. 71 | DOI:10.2478/caim-2019-0013
  • M. Amar; D. Andreucci; R. Gianni Exponential decay for a nonlinear model for electrical conduction in biological tissues, Nonlinear Analysis, Volume 131 (2016), p. 206 | DOI:10.1016/j.na.2015.07.002
  • M. AMAR; D. ANDREUCCI; P. BISEGNA; R. GIANNI Exponential asymptotic stability for an elliptic equation with memory arising in electrical conduction in biological tissues, European Journal of Applied Mathematics, Volume 20 (2009) no. 5, p. 431 | DOI:10.1017/s0956792509990052
  • Micol Amar; Daniele Andreucci; Paolo Bisegna; Roberto Gianni Stability and memory effects in a homogenized model governing the electrical conduction in biological tissues, Journal of Mechanics of Materials and Structures, Volume 4 (2009) no. 2, p. 211 | DOI:10.2140/jomms.2009.4.211
  • Marco Veneroni Reaction–diffusion systems for the macroscopic bidomain model of the cardiac electric field, Nonlinear Analysis: Real World Applications, Volume 10 (2009) no. 2, p. 849 | DOI:10.1016/j.nonrwa.2007.11.008
  • Micol Amar; Daniele Andreucci; Paolo Bisegna; Roberto Gianni Applications of homogenization techniques to the electrical conduction in biological tissues, PAMM, Volume 7 (2007) no. 1, p. 2010013 | DOI:10.1002/pamm.200700041
  • Marco Veneroni Reaction–diffusion systems for the microscopic cellular model of the cardiac electric field, Mathematical Methods in the Applied Sciences, Volume 29 (2006) no. 14, p. 1631 | DOI:10.1002/mma.740
  • MICOL AMAR; DANIELE ANDREUCCI; ROBERTO GIANNI; PAOLO BISEGNA EVOLUTION AND MEMORY EFFECTS IN THE HOMOGENIZATION LIMIT FOR ELECTRICAL CONDUCTION IN BIOLOGICAL TISSUES, Mathematical Models and Methods in Applied Sciences, Volume 14 (2004) no. 09, p. 1261 | DOI:10.1142/s0218202504003623

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