Abstract

We prove that the geodesic equations of all Sobolev metrics of fractional order one and higher on spaces of diffeomorphisms and, more generally, immersions are locally well posed. This result builds on the recently established real analytic dependence of fractional Laplacians on the underlying Riemannian metric. It extends several previous results and applies to a wide range of variational partial differential equations, including the well-known Euler–Arnold equations on diffeomorphism groups as well as the geodesic equations on spaces of manifold-valued curves and surfaces.

中文翻译：

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关于沉浸空间的分数Sobolev度量

摘要

我们证明了在微分形空间上所有分数阶及更高分数的所有Sobolev度量的测地方程在局部上具有较好的位置。此结果建立在最近建立的分数拉普拉斯算子对基础黎曼度量的实际解析依赖性上。它扩展了先前的一些结果，并适用于广泛的变分偏微分方程，包括著名的关于亚同形群的Euler-Arnold方程以及流形值曲线和曲面空间上的测地线方程。