Partial Differential Equations
An elliptic equation with history
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.

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Accepted:
Published online:
DOI: 10.1016/j.crma.2004.02.008
Amar, Micol 1; Andreucci, Daniele 1; Bisegna, Paolo 2; Gianni, Roberto 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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Amar, Micol; Andreucci, Daniele; Bisegna, Paolo; Gianni, Roberto. An elliptic equation with history. Comptes Rendus. Mathématique, Volume 338 (2004) no. 8, pp. 595-598. doi : 10.1016/j.crma.2004.02.008. http://www.numdam.org/articles/10.1016/j.crma.2004.02.008/

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

[2] Amar, M.; Andreucci, D.; Bisegna, P.; Gianni, R. 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] Fabrizio, M. An existence and uniqueness theorem in quasi-static viscoelasticity, Quart. Appl. Math., Volume 47 (1989), pp. 1-9

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

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

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

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