Size estimation of inclusion

Abstract

This paper concerns the estimation of the size(Lebesgue measure) of an unknown inclusion imbedded in a known reference conductor or elasctic body. First, we point out two improvements of the esatimate obtained by Kang H., Seo J. and Sheen D. in their paper "The inverse conductivity problem with one measurement: Stability and Estimation of Size", to appear in SIAM J. Math. Anal.. Second, we establish a system of integral inequalities in the similar problem for an anisotropic elastic body and get similar estimates. AN INEQUALITY BY KANG, SEO AND SHEENLet Ω be a bounded domain of R n (n > 2) with Lipschitz boundary and D a subdomain compactly contained in Ω. Let k be a positive number such that k^l.Direct Problem Find u such that r\ n V · <yVu = 0 in Ω, -^ = g on <9Ω, u = 0°v Jan where g Ε Ζ/ 2 (<9Ω) with / ΟΩ g = 0 and 7 = l + (fc -I)XD- Neumann-to-Dirichlet mapInverse Problem k is known. Determine D from g and Λ^(0) (for finitely many g).There are several works on this subject. Friedman-Isakov [4] proved a covex polyhedron can be determined from one observation data under the distance condition diam(^) < dist (D,dtl) and if Ω is a half space, the assertion is valid even if this distance condition is dropped or if D is a disk. Isakov-Powell [7] extended this resut to a union of disks in a half space.

Probing for electrical inclusions with complex spherical waves

“…Our starting points are the following inequalities given in [39] (see also [20]). LEMMA 2.6 Let f ∈ H 3/2 (∂ ), and u be a solution to the boundary value problem…”

Detecting an Inclusion in an Elastic Body by Boundary Measurements

“…See also Alessandrini and Di Cristo [ADC04] for a positive result in this direction. Here, following the line of research initiated in Alessandrini and Rosset [AR98], Kang, Seo, and Sheen [KSS97], Alessandrini, Rosset, and Seo [ARS00], and Alessandrini and Rosset [AR04] in the electrostatic setting, and in Ikehata [Ik98] for elasticity, we pose a relatively modest but realistic goal: Can we estimate the size (i.e., volume) of the unknown inclusion D from one set of boundary measurements of traction and displacement?…”

Numerical boundary identification for Helmholtz-type equations

“…An important class of inverse problems in engineering consists of the inverse geometrical problems which can be divided into the following subclasses: shape and design optimisation [3][4][5][6][7], identification of defects, e.g. cracks, cavities or inclusions, [8][9][10][11][12][13][14] and identification of an unknown boundary [15][16][17][18][19][20][21][22][23][24].…”