Radiation conditions and uniqueness theorem for n-dimensional wave equation in an infinite domain

Abstract

The present work is intended to be a comprehensive and systematic treatment of the “radiation condition” (a particular case being Sommerfeld's radiation condition) which guarantees the uniqueness of the solution of the exterior boundary value problems for the second-order linear elliptic differential equation (which one can also consider as the reduced general wave equation) L(u) = Σ i,j=1 n a ij(x) ∂ 2u ∂x i ∂x j + ∫ i=1 n b i(x) ∂u ∂x i + c(x)u = 0 in n-dimensional Euclidean space E n . First of all, Sobolev's integral formula is generalized. This is accomplished by means of the concept of retarded argument and auxiliary functions σ n and τ (in an appendix). Furthermore, some additional restrictions are imposed on σ n and τ. Second, using this generalized integral formula, conditions which are a generalization of the classical Sommerfeld's radiation condition are found. Then the maximum principle for the solution in an unbounded domain is stated which finally leads to the uniqueness theorem for the exterior boundary value problem. Special cases of (A) such as Δu + k 2 u = 0 and Δu + k 2( x) u = 0 can also be deduced.