Applications of the Oscillating-Decaying Solutions to Inverse Problems

Ikehata [2] had introduced a method called enclosing method. This is the reconstruction procedure for indentifying the convex hulls of a polygonal or polyhedral set D of the characteristic function χ D of a source term ρ(x)χ D (x) for the Poisson equation Δu = ρχ D in a domain Ω ⊂ ℝ n (n = 2 or 3), a polygonal inclusion D of the conductivity equation ∇ · ((1 + kχ D )u) = 0 in Ω ⊂ ℝ2 and a polygonal cavity D for the Laplace equation Δ u = 0 in $$\Omega \backslash \bar{D} \subset {{\mathbb{R}}^{2}}$$ from any one pair of Cauchy data (f = u|əΩ ≢ const., $$g = \frac{{\partial u}}{{\partial v}}$$ ) under some condition on D. Moreover, this method also works for reconstructing the convex hull of an inclusion D which doesn’t have to be a polygon from many Cauchy data.