Partial Regularity for Harmonic Maps into Spheres at a Singular or Degenerate Free Boundary

TL;DR

This paper establishes partial regularity results for harmonic maps into spheres with free boundaries, even when the domain metric degenerates or becomes singular near the boundary, advancing understanding of singularities in geometric analysis.

Contribution

It introduces new techniques to prove partial regularity for harmonic maps with degenerate or singular metrics at free boundaries, extending previous regularity results.

Findings

Proves partial regularity of harmonic maps with free boundary conditions.

Handles metrics degenerating at the boundary at rate d^α.

Provides new methods for analyzing singularities in geometric PDEs.

Abstract

We prove partial regularity of weakly stationary harmonic maps with (partially) free boundary data on manifolds where the domain metric may degenerate or become singular along the free boundary at the rate $d^\alpha$ for the distance function $d$ from the boundary.