Scattering by Obstacles

Uniqueness of the solution to inverse obstacle scattering problem

“…This result is proved in [1], p.87. The boundary conditions in Ramm's uniqueness theorem could be the Dirichlet, or the Neumann, or the Robin condition.…”

“…Let D ⊂ R 3 be a bounded domain with a smooth (say, twice continuously differentiable), but not analytic, boundary S, (∇ 2 + k 2 )u = 0 in D := R 3 \ D, k = const > 0; u = 0 on S, [1] u = exp(ikα · x) + A(α , α, k)r −1 exp(ikr) + o(r −1 ), r := |x| → ∞, α := xr −1 .…”

“…It is known [1] that problem (1)- (2) has a unique solution, called the scattering solution. The inverse obstacle scattering problem (IP) is: Given A(α , α, k) on a subset of S 2 × S 2 × [0, ∞) find S. The first uniqueness result for IP was proved in 1964 by M.Schiffer.…”

“…In [1], p.86, the result of M.Schiffer is proved and the Dirichlet boundary condition is not assumed a priori, but it is proved that the boundary condition is uniquely determined by the above scattering data.…”

Electromagnetic wave scattering by many small particles

“…If k 2 is such an eigenvalue, then this integral equation can be modified so that the modified equation is uniquely solvable for j (see [8]). If j is found, then formula (11) allows one to calculate U = E H in a neighborhood of S j and on S j .…”