Boundary regularity for conformally invariant variational problems with Neumann data

Abstract

We study boundary regularity of maps from two-dimensional domains into manifolds which are critical with respect to a generic conformally invariant variational functional and which, at the boundary, enter perpendicularly into a support manifold. For example, harmonic maps, or $H$-surfaces, with a partially free boundary condition. In the interior it is known, by the celebrated work of Riviere, that these maps satisfy a system with an antisymmetric potential, from which one can derive regularity of the solution. We show that these maps satisfy along the boundary a system with a nonlocal antisymmetric boundary potential which contains information from the interior potential and the geometric Neumann boundary condition. We then proceed to show boundary regularity for solutions to such systems.

References in corpus (1)

• n/p-Harmonic maps: regularity for the sphere case

Cited by in corpus (4)

• Calderon-Zygmund type estimates for nonlocal PDE with Hölder continuous kernel
• On regularity theory for n/p-harmonic maps into manifolds
• A Resolution of the Poisson Problem for Elastic Plates
• Epsilon-regularity for p-harmonic maps at a free boundary on a sphere