An elliptic equation with history

Micol Amar; Daniele Andreucci; Paolo Bisegna; Roberto Gianni

doi:10.1016/j.crma.2004.02.008

Partial Differential Equations

An elliptic equation with history

Presented by: Philippe G. Ciarlet

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

Comptes Rendus. Mathématique, Volume 338 (2004) no. 8, pp. 595-598.

Abstract
Résumé

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.

Received: 2003-08-25
Accepted: 2004-02-16
Published online: 2004-03-25

DOI: 10.1016/j.crma.2004.02.008

Author's affiliations:

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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     author = {Micol Amar and Daniele Andreucci and Paolo Bisegna and Roberto Gianni},
     title = {An elliptic equation with history},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {595--598},
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     year = {2004},
     doi = {10.1016/j.crma.2004.02.008},
     language = {en},
}

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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/

References
Cited by

[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

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