We consider the problem of minimizing the weighted Dirichlet energy between homeomorphisms of planar annuli. A known challenge lies in the case when the weight λ depends on the independent variable z. We prove that for an increasing radial weight λ(z) the infimal energy within the class of all Sobolev homeomorphisms is the same as in the class of radially symmetric maps. For a general radial weight λ(z) we establish the same result in the case when the target is conformally thin compared to the domain. Fixing the admissible homeomorphisms on the outer boundary we establish the radial symmetry for every such weight.

中文翻译：

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最小化器对加权狄利克雷能量的径向对称性

我们考虑最小化平面环的同胚之间的加权狄利克雷能量的问题。一个已知的挑战在于当重量λ取决于自变量z. 我们证明对于增加的径向重量λ(z) 所有 Sobolev 同胚类中的小能量与径向对称映射类中的相同。对于一般径向重量λ(z) 我们在目标与域相比保形薄的情况下建立了相同的结果。将允许的同胚固定在外边界上，我们为每个这样的权重建立径向对称。