浸入空间上的分数阶 Sobolev 度量
Fractional Sobolev metrics on spaces of immersions
AI总结：
本文利用分数阶 Laplace 算子对底层度量的实解析依赖性，证明微分同胚与浸入空间上一阶及以上分数阶 Sobolev 度量的测地线方程局部适定，并推广至 Euler-Arnold 方程等多种变分 PDE。
摘要：
我们证明了微分同胚空间以及更一般的浸入空间上，所有阶数为一及更高阶的分数阶 Sobolev 度量的测地线方程均局部适定。该结果建立在最近确立的分数阶 Laplace 算子对底层 Riemannian 度量的实解析依赖性之上。它推广了此前若干结果，并适用于广泛的变分偏微分方程，包括微分同胚群上著名的 Euler-Arnold 方程，以及流形值曲线和曲面空间上的测地线方程。
英文摘要：
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.
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