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.

中文翻译：

---

简谐振动问题的椭圆方程

在2D等离子体中谐振波的建模导致两个具有光滑系数$ \ alpha $和紧实项的退化椭圆方程的耦合。系数$ \ alpha $更改符号。$ \ {\ alpha> 0 \} $是传播区域，而$ \ {\ alpha <0 \} $是非传播区域和椭圆区域。这两个模型通过线$ \ varSigma = \ {\ alpha = 0 \} $耦合。通常，这是一个不适定的问题，必须引入其他信息才能以$ \ varSigma $获得满意的治疗。在这项工作中，我们通过在自适应功能设置中依靠限制吸收原理（$ \ alpha $被$ \ alpha + i0 ^ + $代替）来定义解决方案。此设置基于正则和奇异部分的解的分解以及准解，该分解的起源为$ \ varSigma $。这导致了一种新的耦合良好的混合变分公式。当我们设计显式拟解时，可以进行数值实验，这说明了该新工具用于数值计算的良好特性。