Limiting absorption principle and well-posedness for the Helmholtz equation with sign changing coefficients

Abstract

In this paper, we investigate the limiting absorption principle associated to and the well-posedness of the Helmholtz equations with sign changing coefficients which are used to model negative index materials. Using the reflecting technique introduced in \cite{Ng-Complementary}, we first derive Cauchy problems from these equations. The limiting absorption principle and the well-posedness are then obtained via various a priori estimates for these Cauchy problems. There approaches are proposed to obtain the a priori estimates. The first one follows from a priori estimates of elliptic systems equipped general complementing boundary conditions due to Agmon, Douglis, and Nirenberg in their classic work \cite{ADNII}. The second approach, which is complement to the first one, is variational and based on the Dirichlet principle. The last approach, which is complement to the second one, is also variational and uses the multiplier technique. As a consequence, we obtain new results on the well-posedness of these equations for which the positivity conditions concerning the matrix-valued coefficients are imposed partially or not strictly on the interface of sign changing coefficients. As a result, the well-posedness can hold even in the case the contrast of the coefficients across the sign changing interfaces is arbitrary. This allows us to rediscover and extend largely known results obtained by the integral method, the pseudo differential operator theory, and the T-coercivity approach. The unique solution, obtained by the limiting absorption principle, is {\bf not} in $H^1_{\loc}(\mR^d)$ as usual and possibly {\bf not even} in $L^2_{\loc}(\mR^d)$. The optimality of our results is also discussed.