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arXiv · 2108.07643

On 2D Harmonic Extensions of Vector Fields and Stellarator Coils

Adam J. Golab·James C. Robinson·José L. Rodrigo
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Abstract

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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Adam J. Golab, James C. Robinson, José L. Rodrigo. 2021-08-17. On 2D Harmonic Extensions of Vector Fields and Stellarator Coils. https://arxiv.org/abs/2108.07643

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