关于向量场和仿星器线圈的二维调和延拓
On 2D Harmonic Extensions of Vector Fields and Stellarator Coils
摘要：
我们考虑一个有关的问题，称为仿星器的磁约束装置。等离子体是由载流线圈产生的磁场限制，在这里，我们研究如何接近等离子体，他们需要定位。载流线圈被表示为磁场内的奇点，因此这个问题可以在数学上建模为找到我们可以从域的边界谐波地延伸向量场多远。本文考虑具有实解析边界的二维区域，证明了调和扩张存在的充要条件是边界数据满足相容性和正则性条件。我们的证明方法使用了推广的结果阿达玛柯西问题的拉普拉斯算子。然后，我们提供了一个下界多远，我们可以通过柯西-Kovalevskaya定理从边界调和扩展的向量场。
英文摘要：
We consider a problem relating to magnetic confinement devices known as stellarators. Plasma is confined by magnetic fields generated by current-carrying coils, and here we investigate how closely to the plasma they need to be positioned. Current-carrying coils are represented as singularities within the magnetic field and therefore this problem can be modelled mathematically as finding how far we can harmonically extend a vector field from the boundary of a domain. For this paper we consider two-dimensional domains with real analytic boundary, and prove that a harmonic extension exists if and only if the boundary data satisfies a combined compatibility and regularity condition. Our method of proof uses a generalisation of a result of Hadamard on the Cauchy problem for the Laplacian. We then provide a lower bound on how far we can harmonically extend the vector field from the boundary via the Cauchy--Kovalevskaya Theorem.
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