Two-Point Boundary Value Problems on Diffeomorphism Groups

TL;DR

This paper studies geodesic equations on Sobolev diffeomorphism groups, proving boundary value solutions are as smooth as boundary data and constructing smooth exponential maps in the Fréchet setting.

Contribution

It establishes the regularity of solutions to two-point boundary value problems and constructs differentiable exponential maps on diffeomorphism groups.

Findings

Solutions are as smooth as boundary conditions.

Constructed differentiable exponential maps.

Applicable to equations of ideal hydrodynamics.

Abstract

We consider a variety of geodesic equations on Sobolev diffeomorphism groups, including the equations of ideal hydrodynamics. We prove that solutions of the corresponding two-point boundary value problems are precisely as smooth as their boundary conditions. We further utilise this regularity property to construct continuously differentiable exponential maps in the Frech\'et setting.