Transmission eigenvalues and a problem of Hans Lewy

Subject to certain assumptions on the matrix N=N(x) we show that except for a discrete set of values of k (called transmission eigenvalues) the only solution of∇·N∇u+k2u=0Δv+k2v=0 in D,u=v∂u∂ν=∂v∂ν on ∂D,where D⊂Rn is a domain with smooth boundary ∂D having unit outward normal ν is the trivial solution u=v=0. This problem has important applications to the inverse scattering problem for anisotropic media and is in a class of problems first considered by Lewy (Bull. Amer. Math. Soc. 65 (1959) 37–58).