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On the linear force-free fields in bounded and unbounded three-dimensional domains. (English) Zbl 0954.35043

Solutions of the nonlinear system of equations \({\mathcal K}:= \text{curl }B\times B= 0\), satisfying \(\text{div }B= 0\), are known as force-free magnetic fields, since the Lorentz force \({\mathcal K}\) of the current generated by the magnetic field \(B\) is assumed to vanish. Particular solutions are found as eigensolutions associated with curl (Beltrami) fields \[ \text{curl }B-\alpha B=0.\tag{FFF} \] The authors consider such fields in bounded domains with smooth boundary, in half-cylinder domains with smooth bounded cross-section, as well as in the exterior of the unit ball. For a bounded domain \(\Omega\) a Fredholm type solution theory for the problem (FFF) with prescribed boundary data \(g= n\cdot B\) on \(\partial\Omega\), \(n\) exterior normal, and prescribed projection on the space \({\mathcal H}_N\) of harmonic Neumann vector fields (in terms of conditions of the form \(\int_{\partial\Omega}(n\times B)\cdot q_ido= \alpha a_i\), \(i= 1,\dots, m\), where \((q_i)_{i= 1,\dots,m}\) is a particular bases of \({\mathcal H}_N)\). Beyond this, however, the question is raised and answered how the parameter \(\alpha\) can be determined from suitable data \(a:=(a_i)_{i= 1,\dots, N}\in \mathbb{R}^m\) and \(g\) given such that the helicity type quantity \[ m_0:= \int_\Omega B\cdot\text{curl } B dV \] is prescribed. In a second part the authors focus on the case that \(\Omega\) is a half-cylinder \(\widetilde\Omega\times ]0,\infty[\) and derive an existence and uniqueness result with \(n\cdot B\equiv e_3\cdot B\) prescribed on the bottom \(\widetilde\Omega\times \{0\}\) of \(\Omega\). On the wall \(\partial\widetilde\Omega\times ]0,\infty[\) the boundary condition \(e_3\cdot B= 0\) is imposed.
In a final part the case of the exterior of the unit ball is considered. This investigation is based on explicit calculations in terms of spherical harmonics.

MSC:

35F15 Boundary value problems for linear first-order PDEs
35Q60 PDEs in connection with optics and electromagnetic theory
78A25 Electromagnetic theory (general)
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