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

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 $$d^\alpha $$ d α for the distance function d from the boundary.

Partial regularity for minimizers of discontinuous quasiconvex integrals with general growth

We prove the partial Hölder continuity for minimizers of quasiconvex functionals \begin{equation*} \mathcal{F}({\bf u}) \colon =\int_{\Omega} f(x,{\bf u},D{\bf u})\,\textrm{d} x, \end{equation*} where $f$ satisfies a uniform VMO condition with respect to the $x$ -variable and is continuous with respect to ${\bf u}$ . The growth condition with respect to the gradient variable is assumed a general one.