Abstract
Let \(F : H^q \rightarrow H^q\) be a \(C^k\)-map between Sobolev spaces, either on \({{\mathbb {R}}}^d\) or on a compact manifold. We show that equivariance of F under the diffeomorphism group allows to trade regularity of F as a nonlinear map for regularity in the image space: for \(0 \le l \le k\), the map \(F: H^{q+l} \rightarrow H^{q+l}\) is well defined and of class \(C^{k-l}\). This result is used to study the regularity of the geodesic boundary value problem for Sobolev metrics on the diffeomorphism group and the space of curves.
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The author was partially supported by the Erwin Schrödinger Institute programme Infinite-Dimensional Riemannian Geometry with Applications to Image Matching and Shape Analysis.
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Bruveris, M. Regularity of maps between Sobolev spaces. Ann Glob Anal Geom 52, 11–24 (2017). https://doi.org/10.1007/s10455-017-9544-6
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DOI: https://doi.org/10.1007/s10455-017-9544-6