Dimensional estimates for singular sets in geometric variational   problems with free boundaries
  We show that singular sets of free boundaries arising in codimension one anisotropic geometric variational problems are $\mathcal H ^{n-3}$-negligible, where $n$ is the ambient space dimension. In particular our results apply to capillarity type problems, and establish everywhere regularity in the three-dimensional case. 
Guido De Philippis and Francesco Maggi
math.AP