Identification of cracks with non linear impedances
ESAIM: Modélisation mathématique et analyse numérique, Tome 37 (2003) no. 2, pp. 241-257.

We consider the inverse problem of determining a crack submitted to a non linear impedance law. Identifiability and local Lipschitz stability results are proved for both the crack and the impedance.

DOI : 10.1051/m2an:2003033
Classification : 35R30, 35J25
Mots clés : inverse problems, cracks
Jaoua, Mohamed  ; Nicaise, Serge 1 ; Paquet, Luc 

1 Université de Valenciennes et du Hainaut Cambrésis, MACS, Le Mont Houy, 59313 Valenciennes Cedex 9, France. http://www.univ-valenciennes.fr/macs/Serge.Nicaise
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     title = {Identification of cracks with non linear impedances},
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Jaoua, Mohamed; Nicaise, Serge; Paquet, Luc. Identification of cracks with non linear impedances. ESAIM: Modélisation mathématique et analyse numérique, Tome 37 (2003) no. 2, pp. 241-257. doi : 10.1051/m2an:2003033. http://www.numdam.org/articles/10.1051/m2an:2003033/

[1] G. Alessandrini, Stability for the crack determination problem, in Inverse problems in Mathematical Physics, L. Päıvaärinta and E. Somersalo Eds., Springer-Verlag, Berlin (1993) 1-8. | Zbl

[2] G. Alessandrini, E. Beretta and S. Vessella, Determining linear cracks by boundary measurements: Lipschitz stability. SIAM J. Math. Anal. 27 (1996) 361-375. | Zbl

[3] G. Alessandrini and A. Diaz Valenzuela, Unique determination of multiple cracks by two measurements. SIAM J. Control Optim. 34 (1996) 913-921. | Zbl

[4] G. Alessandrini and A. Dibenedetto, Determining 2-dimensional cracks in 3-dimensional bodies: uniqueness and stability. Indiana Univ. Math. J. 46 (1997) 1-82. | Zbl

[5] S. Andrieux and A. Ben Abda, Identification of planar cracks by overdetermined boundary data: inversion formulae. Inverse Problems 12 (1996) 553-563. | Zbl

[6] S. Andrieux, A. Ben Abda and M. Jaoua, On the inverse emerging plane crack problem. Math. Methods Appl. Sci. 21 (1998) 895-907. | Zbl

[7] A. Ben Abda, H. Ben Ameur and M. Jaoua, A semi-explicit algorithm for the reconstruction of 3D planar cracks. Inverse Problems 15 (1999) 67-78. | Zbl

[8] R. Bellout and A. Friedman, Identification problems in potential theory. Arch. Rational Mech. Anal. 101 (1988) 143-160. | Zbl

[9] M. Bonnet, Boundary Integral Equation Methods for Solids and Fluids. Wiley, New York (1995). | Zbl

[10] K. Bryan and M. Vogelius, A uniqueness result concerning the identification of a collection of cracks from finitely many electrostatic boundary measurements. SIAM J. Math. Anal. 23 (1992) 950-958. | Zbl

[11] M. Dauge, Elliptic boundary value problems in corner domains. Smoothness and asymptotics of solutions. Springer Verlag, Berlin, Lecture Notes in Math. 1341 (1988). | MR | Zbl

[12] C. Dellacherie and P.-A. Meyer, Probabilité et potentiel. Hermann (1975). | MR | Zbl

[13] P. Destuynder and M. Jaoua, Sur une interprétation mathématique de l'intégrale de Rice en théorie de la rupture fragile. Math. Methods Appl. Sci. 3 (1981) 70-87. | Zbl

[14] R. Felfel, Étude de l'identifiabilité et de la stabilité d'une fissure présentant une résistivité de contact. DEA de Mathématiques Appliquées, ENIT, Tunis (1997).

[15] A. Friedman and M. Vogelius, Determining cracks by boundary measurements. Indiana Univ. Math. J. 38 (1989) 527-556. | Zbl

[16] P. Grisvard, Elliptic problems in nonsmooth domains. Pitman, Boston (1985). | MR | Zbl

[17] V.G. Maz'Ya and B.A. Plamenevsky, On the coefficients in the asymptotics of solutions of elliptic boundary value problems in domains with conical points. Amer. Math. Soc. Transl. Ser. 2 123 (1984) 57-88. | Zbl

[18] F. Murat and J. Simon, Quelques résultats sur le contrôle par un domaine géométrique. Preprint, Université de Paris VI (1974). | MR

[19] E.P. Stephan, Boundary integral equations for mixed boundary value problems, screen and transmission problems in 3 . Habilitationsschrift, TH Darmstadt, Germany (1984).

[20] E.P. Stephan, Boundary integral equations for screen problems in 3 . Integral Equations Operator Theory 10 (1987) 236-257. | Zbl

[21] V.S. Vladimirov, Equations of Mathematical Physics. Marcel Dekker, New York (1971). | MR | Zbl

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