We study the possibility of constructing steady magnetic fields satisfying the force balance equation of ideal magnetohydrodynamics with tangential boundary conditions in asymmetric confinement vessels, i.e. bounded regions that are not invariant under continuous Euclidean isometries (translations, rotations, or their combination). This problem is often encountered in the design of next-generation fusion reactors. We show that such configurations are possible if one relaxes the standard assumption that the vessel boundary corresponds to a pressure isosurface. We exhibit a smooth solution that possesses an Euclidean symmetry and yet solves the boundary value problem in an asymmetric ellipsoidal domain while sustaining a non-vanishing pressure gradient. This result provides a definitive answer to the problem of existence of regular ideal magnetofluidostatic equilibria in asymmetric bounded domains. The question remains open whether regular asymmetric solutions of the boundary value problem exist.

中文翻译：

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非对称约束容器中具有非零压力梯度的对称理想磁流体静力平衡

我们研究了在非对称约束容器中构建满足具有切向边界条件的理想磁流体力学的力平衡方程的稳定磁场的可能性，即在连续欧几里德等距（平移、旋转或它们的组合）下不不变的有界区域。这个问题在下一代聚变反应堆的设计中经常遇到。我们表明，如果放宽容器边界对应于压力等值面的标准假设，则这种配置是可能的。我们展示了一个平滑的解决方案，它具有欧几里德对称性，并且在保持非零压力梯度的同时解决了非对称椭球域中的边界值问题。该结果为在非对称有界域中存在规则的理想磁流体平衡的问题提供了明确的答案。边值问题的正则非对称解是否存在，这个问题仍然悬而未决。