Unique continuation from Cauchy data in unknown non-smooth domains

Rondi, Luca

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 5 (2006) no. 2, pp. 189-218.

We consider a conducting body which presents some (unknown) perfectly insulating defects, such as cracks or cavities, for instance. We perform measurements of current and voltage type on a (known) part of the boundary of the conductor. We prove that, even if the defects are unknown, the current and voltage measurements at the boundary uniquely determine the corresponding electrostatic potential inside the conductor. A corresponding stability result, related to the stability of Neumann problems with respect to domain variations, is also proved. Some applications of these results to inverse problems are presented.

[1] G. Alessandrini, E. Beretta, E. Rosset and S. Vessella, Optimal stability for inverse elliptic boundary-value problems with unknown boundaries, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 29 (2000), 755-806. | Numdam | MR | Zbl

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

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

[4] L. Ambrosio, N. Fusco and D. Pallara, “Functions of Bounded Variation and Free Discontinuity Problems”, Clarendon Press, Oxford, 2000. | MR | Zbl

[5] K. Bryan and M. S. Vogelius, A review of selected works on crack identification, In: “Geometric Methods in Inverse Problems and PDE Control”, C. B. Croke, I. Lasiecka, G. Uhlmann and M. S. Vogelius (eds.), Springer-Verlag, New York, 2004, 25-46. | MR | Zbl

[6] G. Dal Maso, F. Ebobisse and M. Ponsiglione, A stability result for nonlinear Neumann problems under boundary variations, J. Math. Pures Appl. 82 (2003), 503-532. | MR | Zbl

[7] J. Deny and J. L. Lions, Les espaces du type de Beppo Levi, Ann. Inst. Fourier (Grenoble) 5 (1953-54), 305-370. | Numdam | MR | Zbl

[8] M. Di Cristo and L. Rondi, Examples of exponential instability for inverse inclusion and scattering problems, Inverse Problems 19 (2003), 685-701. | MR | Zbl

[9] L. C. Evans and R. F. Gariepy, “Measure Theory and Fine Properties of Functions”, CRC Press, Boca Raton Ann Arbor London, 1992. | MR | Zbl

[10] H. Federer, “Geometric Measure Theory”, Springer-Verlag, Berlin Heidelberg New York, 1969. | MR | Zbl

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

[12] A. Giacomini, A stability result for Neumann problems in dimension $N\ge 3$, J. Convex Anal. 11 (2004), 41-58. | MR | Zbl

[13] E. Giusti, “Minimal Surfaces and Functions of Bounded Variation”, Birkhäuser, Boston Basel Stuttgart, 1984. | MR | Zbl

[14] J. Heinonen, T. Kilpeläinen and O. Martio, “Nonlinear Potential Theory of Degenerate Elliptic Equations”, Clarendon Press, Oxford New York Tokyo, 1993. | MR | Zbl

[15] D. S. Jerison and C. E. Kenig, The Neumann problem on Lipschitz domains, Bull. Amer. Math. Soc. (N.S.) 4 (1981), 203-207. | MR | Zbl

[16] V. G. Maz'Ja, “Sobolev Spaces”, Springer-Verlag, Berlin Heidelberg New York, 1985. | MR

[17] N.G. Meyers, An $L^p$-estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 17 (1963), 189-206. | Numdam | MR | Zbl

[18] D. Mumford and J. Shah, Boundary detection by minimizing functionals, I, In: “Proceedings of IEEE Computer Society Conference on Computer Vision and Pattern Recognition”, IEEE Computer Society Press/North-Holland, Silver Spring Md./Amsterdam, 1985, 22-26.

[19] D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems, Comm. Pure Appl. Math. 42 (1989), 577-685. | MR | Zbl

[20] M. K. V. Murthy and G. Stampacchia, A variational inequality with mixed boundary conditions, Israel J. Math. 13 (1972), 188-224. | MR | Zbl

[21] L. Rondi, Uniqueness and Optimal Stability for the Determination of Multiple Defects by Electrostatic Measurements, Ph.D. thesis, S.I.S.S.A.-I.S.A.S., Trieste, 1999 (downloadable from http://www.sissa.it/library/).

[22] L. Rondi, Optimal stability of reconstruction of plane Lipschitz cracks, SIAM J. Math. Anal. 36 (2005), 1282-1292. | MR | Zbl

[23] L. Rondi and F. Santosa, Enhanced Electrical Impedance Tomography via the Mumford-Shah Functional, ESAIM: COCV 6 (2001), 517-538. | Numdam | MR | Zbl