Regularity of Maps between Sobolev Spaces

2016/02/21 by Bruveris, Martins · 1 citation
#58B20 #Differential Geometry (math.DG) #FOS: Mathematics #Primary 58D15 #Secondary 58D05

paper · doi:10.48550/arxiv.1602.06558

Abstract

Let F : H^q → 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 ≤ l ≤ k, the map F: H^q+l → 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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