We address the existence of stationary solutions of the Vlasov-Poisson system on a domain $ \Omega\subset\mathbb{R}^3 $ describing a high-temperature plasma which due to the influence of an external magnetic field is spatially confined to a subregion of $ \Omega $. In a first part we provide such an existence result for a generalized system of Vlasov-Poisson type and investigate the relation between the strength of the external magnetic field, the sharpness of the confinement and the amount of plasma that is confined measured in terms of the total charges. The key tools here are the method of sub-/supersolutions and the use of first integrals in combination with cutoff functions. In a second part we apply these general results to the usual Vlasov-Poisson equation in three different settings: the infinite and finite cylinder, as well as domains with toroidal symmetry. This way we prove the existence of stationary solutions corresponding to a two-component plasma confined in a Mirror trap, as well as a Tokamak.

中文翻译：

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限制静态Vlasov-Poisson等离子体构型的一般方法

我们讨论了Vlasov-Poisson系统在域\ \ Omega \ subset \ mathbb {R} ^ 3 $上的平稳解的存在，该域描述了高温等离子体，该等离子体由于外部磁场的影响而在空间上局限于一个$ \ Omega $。在第一部分中，我们为Vlasov-Poisson型广义系统提供了这样的存在结果，并研究了外部磁场强度，限制的锐度和被限制的等离子量之间的关系，该关系由总费用。这里的关键工具是子/超解法以及结合积分函数使用第一积分的方法。在第二部分中，我们将这些一般结果应用于三种不同设置下的常规Vlasov-Poisson方程：无限圆柱和有限圆柱体，以及具有环形对称性的域。通过这种方式，我们证明了对应于限定在Mirror阱和Tokamak中的两组分等离子体的平稳溶液的存在。