Domain Identification for Semilinear Elliptic Equations in the Plane: the Zero Flux Case

Abstract

We consider the problem of identifying the domain ci C R2 of a semilinear elliptic equation subject to given Cauchy data on part of the known outer boundary I' and to the zero flux condition on the unknown inner boundary y, where it is assumed that r is a piecewise C' curve and that y is the boundary of a finite disjoint union of simply connected domains, each bounded by a piecewiseC' Jordan curve. It is shown that, under appropriate smoothness conditions, the domain ci is uniquely determined. The problem of existence of solution for given data is not considered since it is usually of lesser importance in view of measurement errors giving data for which no solution exists.

Identification of Cavities in a Three Dimensional Elastic Body

“…In the present paper, we study cavities having a non-smooth boundary, and the problem is associated to the three dimensional Lamé system. Our paper can be seen as a direct extension of [25]. In fact, from the point of view of fracture mechanics, the surfaces of cavities often have cones, sharp edges etc.…”

“…e.g. [10,24,25,26]). For the determination of irregular cavities in a two dimensional elastic body, we refer to [12,25].…”

“…[10,24,25,26]). For the determination of irregular cavities in a two dimensional elastic body, we refer to [12,25]. We also refer to [6,8,27] (and the references therein) for the problem related to cavities in elastic bodies and for the unique continuation for the Lamé system.…”

Cavity detection by the electric method: The 3‐dimensional case

“…In [24], L. Rondi proved that, by assigning a suitable current density on the subarc Γ 0 , a measurement of the corresponding potential on Γ 0 determines uniquely the non-smooth inner boundary. A 2-dimensional analogue of Theorem 1 for non-smooth γ was proved in [28].…”

“…), two measurements are needed for uniqueness in the case of rectilinear cracks. The identification problem for regular cavities using the electric method was considered in [3,8,11,[13][14][15][16][17]19,20,27,28] and others. In some of the references mentioned above, e.g., [8,19,20], the problem of approximating the solution is also considered, whereas in others, e.g., [3,[13][14][15][16][17], the stability problem is studied.…”