Degenerate elliptic equations for resonant wave problems

Abstract

Abstract The modelling of resonant waves in 2D plasma leads to the coupling of two degenerate elliptic equations with a smooth coefficient $\alpha $ and compact terms. The coefficient $\alpha $ changes sign. The region where $\{\alpha>0\}$ is propagative, and the region where $\{\alpha <0\}$ is non propagative and elliptic. The two models are coupled through the line $\varSigma =\{\alpha =0\}$. Generically, it is an ill-posed problem and additional information must be introduced to get a satisfactory treatment at $\varSigma $. In this work, we define the solution by relying on the limiting absorption principle ($\alpha $ is replaced by $\alpha +i0^+$) in an adapted functional setting. This setting lies on the decomposition of the solution in a regular and a singular part, which originates at $\varSigma $, and on quasi-solutions. It leads to a new well-posed mixed variational formulation with coupling. As we design explicit quasi-solutions, numerical experiments can be carried out, which illustrate the good properties of this new tool for numerical computation.