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We consider a problem relating to magnetic confinement devices known asstellarators. Plasma is confined by magnetic fields generated bycurrent-carrying coils, and here we investigate how closely to the plasma theyneed to be positioned. Current-carrying coils are represented as singularitieswithin the magnetic field and therefore this problem can be modelledmathematically as finding how far we can harmonically extend a vector fieldfrom the boundary of a domain.  For this paper we consider two-dimensional domains with real analyticboundary, and prove that a harmonic extension exists if and only if theboundary data satisfies a combined compatibility and regularity condition. Ourmethod of proof uses a generalisation of a result of Hadamard on the Cauchyproblem for the Laplacian. We then provide a lower bound on how far we canharmonically extend the vector field from the boundary via theCauchy--Kovalevskaya Theorem.

On 2D Harmonic Extensions of Vector Fields and Stellarator Coils