Hausdorff-dimension upper bound for the singular set

Establish an upper bound for the Hausdorff dimension of the singular set of the free boundary associated with the two-species long-range segregation system in dimensions higher than two.

Background

The paper studies a two-species elliptic segregation system with nonlocal interactions in an annular domain and analyzes the geometry of its free boundaries. Prior work established regularity results and produced two higher-dimensional examples whose singular sets have Hausdorff dimension exactly n-2.

A general upper bound for the Hausdorff dimension of the singular set in higher dimensions is not established in the cited literature. Resolving this problem would extend the known partial regularity theory beyond the specific examples discussed in the paper.