Exterior Boundary Value Problems as Limits of Interface Problems

It is proved that the solution to exterior Neumann boundary value problem can be obtained as the limit of the solutions of some problems in the whole space. 1 Consider the following problem: (V' + k*)u =f in f2, ul,=O, where f2 = R3\D. D is bounded domain with a smooth boundary r, fE C,(Q), k > 0. In [l] we proved the following: THEOREM 1. Consider the problem V=N in D N = const. > 0. =0 in Q, Then us-+ u as N + +CO and u is the solution of (1). Convergence is understood in the norm ofL'(f2; (1 + IxI)-('+')), E > 0. For convenience of the reader we give a short proof in Appendix 2. The proof differs from the proof given later by Lax and Phillips [4]. The purpose of this paper is to prove similar theorem for the Neumann boundary value problems (bvp). 256 257 2 Consider the bvp (V' + k2)U =-f in R, (V2+k2)u=0 in D, (3) au+ yu+=u-,hF= where + (denote te the limits from inside (outside) of D, n denotes the outward pointing unit normal to r, y, h are positive constants. LEMMA 1. Problem (3)-(4) has at most one solution.