二维Matkowski--Sutô方程及其全纯与严格递增生成元
The two-dimensional Matkowski--Sutô equation with holomorphic and strictly increasing generators
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中文总结 AI 辅助
本文研究二维Matkowski--Sutô方程，在全纯与单调算子两种设定下刻画解，全纯情形仅有仿射与指数对，单调情形存在无限维非仿射解。
中文摘要 AI 辅助
我们研究了二维Matkowski--Sutô方程，该方程要求两个拟算术平均数的和等于算术平均数的两倍，并在两种设定下进行探讨。对于复平面凸域上具有凸像的全纯单射生成元，其解恰好是仿射对和具有非零复指数的指数对，且生成元可作仿射变换。可容许的指数依赖于域的形状，并由其边界的曲率准则描述。在Tóth的单调算子框架中，我们构造了全平面上无限维的非仿射剪切对族。它们的生成元在单调算子意义下严格递增，且不必可微。这些对满足任意变量数的加权方程。一维问题的刚性，归功于Daróczy和Páles，在全纯性下依然成立，但在单调性下则不成立。
英文摘要
We study the two-dimensional Matkowski--Sutô equation, which asks for two quasi-arithmetic means whose sum is twice the arithmetic mean, in two settings. For holomorphic injective generators with convex images on a convex domain in the complex plane, the solutions are exactly the affine pairs and the exponential pairs with a nonzero complex exponent, up to affine changes of the generators. The admissible exponents depend on the shape of the domain and are described by a curvature criterion for its boundary. In the monotone-operator framework of Tóth, we construct an infinite-dimensional family of non-affine shear pairs on the whole plane. Their generators are strictly increasing in the sense of monotone operators and need not be differentiable. These pairs solve the weighted equation for any number of variables. The rigidity of the one-dimensional problem, due to Daróczy and Páles, persists under holomorphy but not under monotonicity.
发表机构
- Shizuoka University（静冈大学）
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