Determination of Density of Elliptic Potential

Abstract

In this paper, using techniques of finding boundary conditions for the volume (Newton) potential, we obtain the boundary conditions for the volume potential u (x) = ∫Ω ε (x, ξ)ρ (ξ) dξ, where ε (x, ξ) is the fundamental solution of the following elliptic equation L(x,D)ε(x, ξ) = −∑ (∂/∂xi) aij(x) (∂/∂xj) ε (x, ξ) + a(x)ε (x, ξ) = δ(x, ξ). Using the explicit boundary conditions for the potential u (x), the density ρ (x) of this potential is uniquely determined. Also, the inverse Sommerfeld problem for the Helmholtz equation is considered.