Degenerate elliptic equations for resonant wave problems

The modelling of resonant waves in 2D plasma leads to the coupling of two degenerate elliptic equations with a smooth coefficient α and compact terms. The coefficient α changes sign. The region where {α > 0} is propagative, and the region where {α < 0} is non propagative and elliptic. The two models are coupled through the line Σ = {α = 0}. Generically, it is an ill-posed problem and additional information must be introduced to get a satisfactory treatment at Σ. In this work, we define the solution by relying on the limiting absorption principle (α is replaced by α + 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 Σ, 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.