Transmission problems for conical and quasi-conical at infinity domains

Let be a smooth unbounded domain in conical at infinity, We consider general transmission problems defined by a differential equation 1 and transmission conditions on the boundary 2 where the coefficients are discontinuous on functions, such that the space of infinitely differentiable functions in bounded with all derivatives, is a jump of the function on We give a criterion for the operatorof the transmission problem (1) and (2) to be Fredholm. We also extend this result to more general quasi-conical at infinity domains. This criterion is applied to the anisotropic acoustic problem 3 where is a uniformly positive definite matrix on with discontinuous on entries such that , is discontinuous on function such that is a conormal derivative. We prove that if the acoustic medium is absorbed at infinity the problem (3) has an unique solution for every