Enclosing a polygonal cavity in a two-dimensional bounded domain from Cauchy data

Abstract

We consider a reconstruction problem of the shape of an unknown open set D in a two-dimensional bounded domain from the Cauchy data on ∂ of a nonconstant solution u of the equation u = 0 in \ D. We assume that the Neumann derivative of u vanishes on ∂D and that D is a convex open polygon. We give a formula for the calculation of the support function of D from such data.

Probing for electrical inclusions with complex spherical waves

“…Kohn and Vogelius show that piecewise analytic conductivities are uniquely determined by the DN map [41]. In some cases it is known that only one boundary measurement is sufficient to guarantee the uniqueness; see [15,21,22,38,43] and the references therein. The reconstruction method presented in this paper is different from these previous studies.…”

Complex geometrical optics solutions for anisotropic equations and applications

“…Using the complex geometrical optics solutions in the reconstruction of embedded objects from boundary measurements was first introduced by Ikehata [11], [12]. He called this method the enclosure method.…”

“…For example, one can reconstruct the convex hull of the object with Calderón type solutions [11], [12]. With complex spherical waves, one may be able to reconstruct some nonconvex parts of the object [8], [19].…”

Enclosure method and reconstruction of a linear crack in an elastic body

“…The enclosure method [2,3,4,5] is a methodology in inverse problems for partial differential equations. The method yields a partial information about the location of unknown discontinuity which appears as discontinuity of the coefficients of a partial differential equation or a part of the boundary of the common domain of definition of solutions of the equation.…”

“…For τ > 0 we define a function v = (ω + iω ⊥ )e τ x·(ω+iω ⊥ ) . In the case of anisotropic body, let ω = ω (1) + Re{λ}ω (2) for given two linearly independent real vectors ω (1) = (ω 2) ) . Here λ is a complex number having a positive imaginary part and a solution of the equation…”