Stability and Uniqueness for the Crack Identification Problem

This paper deals with the identifiability of nonsmooth defects by boundary measurements, and the stability of their detection. We introduce and analyze a new pointwise regularity concept at the boundary of an open set which turns out to play a crucial role in the identifiability of defects by two boundary measurements. As a consequence, we prove the unique identifiability for a large class of closed sets, including sets with an infinite number of connected components of positive capacity and totally disconnected sets. In order to rigorously justify numerical approximation results of defects by optimal design methods, we prove a geometric stability result of the defect identification process, without any a priori smoothness assumptions.

[1]  L. Hedberg,et al.  Spectral synthesis in sobolev spaces, and uniqueness of solutions of the Dirichlet problem , 1981 .

[2]  Avner Friedman,et al.  Determining Cracks by Boundary Measurements , 1989 .

[3]  Enrique Zuazua,et al.  Approximation par éléments finis de problèmes elliptiques d'optimisation de forme , 2004 .

[4]  Jin Keun Seo,et al.  Unique determination of a collection of a finite number of cracks from two boundary measurements , 1996 .

[5]  Gianni Dal Maso,et al.  Some necessary and sufficient conditions for the convergence of sequences of unilateral convex sets , 1985 .

[6]  A. N. Tikhonov,et al.  Solutions of ill-posed problems , 1977 .

[7]  Luca Rondi,et al.  Optimal Stability for the Inverse Problemof Multiple Cavities , 2001 .

[8]  Dorin Bucur.  Characterization for the Kuratowski Limits of a Sequence of Sobolev Spaces , 1999 .

[9]  Enrique Zuazua,et al.  Finite Element Approximation of 2D Elliptic Optimal Design , 2006 .

[10]  Marcello Ponsiglione,et al.  A STABILITY RESULT FOR NONLINEAR NEUMANN PROBLEMS UNDER BOUNDARY VARIATIONS , 2002 .

[11]  J. Heinonen,et al.  Nonlinear Potential Theory of Degenerate Elliptic Equations , 1993 .

[12]  Kilpeläinen Tero,et al.  Supersolutions to Degenerate Elliptic Equations on Quasi open Sets , 1992 .

[13]  Kurt Bryan,et al.  A REVIEW OF SELECTED WORKS ON CRACK IDENTIFICATION , 2004 .

[14]  Luca Rondi,et al.  Optimal Stability of Reconstruction of Plane Lipschitz Cracks , 2005, SIAM J. Math. Anal..

[15]  V. Sverák,et al.  On optimal shape design , 1992 .

[16]  DORIN BUCUR,et al.  A Duality Approach for the Boundary Variation of Neumann Problems , 2002, SIAM J. Math. Anal..

[17]  Vivette Girault,et al.  Finite Element Methods for Navier-Stokes Equations - Theory and Algorithms , 1986, Springer Series in Computational Mathematics.

[18]  H. Brezis.  Analyse fonctionnelle : théorie et applications , 1983 .

[19]  Dorin Bucur,et al.  N-Dimensional Shape Optimization under Capacitary Constraint , 1995 .

[20]  Giovanni Alessandrini,et al.  Unique Determination of Multiple Cracks by Two Measurements , 1996 .