This note deals with the existence and uniqueness of a minimiser of the following Grötzsch type problem $$\begin{aligned} \inf _{f\in {\mathcal {F}}}\mathop {\iint }\limits _{Q_{1}}\psi ({\mathbb {K}}(z,f))dxdy \end{aligned}$$ inf f ∈ F ∬ Q 1 ψ ( K ( z , f ) ) d x d y under some conditions, where $${\mathcal {F}}$$ F denotes the set of all orientation preserving homeomorphisms f with the boundary correspondence. As an application, we consider a similar problem of Nitsche type, which concerns the minimiser of a weighted functional for mappings between two annuli with boundary values given by a spiral-stretch map. Furthermore, we give a sharp integral estimate for the rotation angle in terms of the distortion function.

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一个 Grötzsch 型极值问题及其应用

本笔记处理以下 Grötzsch 类型问题的最小化器的存在性和唯一性 $$\begin{aligned} \inf _{f\in {\mathcal {F}}}\mathop {\iint }\limits _{Q_ {1}}\psi ({\mathbb {K}}(z,f))dxdy \end{aligned}$$ inf f ∈ F ∬ Q 1 ψ ( K ( z , f ) ) dxdy 在某些条件下，其中$${\mathcal {F}}$$ F 表示具有边界对应的所有方向保持同胚 f 的集合。作为一个应用，我们考虑一个类似的 Nitsche 类型问题，它涉及两个环之间映射的加权函数的最小化，其边界值由螺旋拉伸图给出。此外，我们根据失真函数对旋转角进行了清晰的积分估计。