Interface distributions in partial segregation problems

We study the minimization of the three-component Dirichlet energy, subject to a partial segregation constraint. A minimizer induces a geometric decomposition of the domain into open regions in which at most two components are active. The active components are harmonic in each region, so the distributional Laplacian of every component is supported on the free boundary, and we analyze the interface Laplacians Λi:=−Δui as signed measures attached to the phase decomposition. Using the regularity and free-boundary theory of Soave and Terracini [On some singularly perturbed elliptic systems modeling partial segregation: uniform Hölder estimates and basic properties of the limits. 2024. Preprint, arXiv:2409.11976v2; On partially segregated harmonic maps: optimal regularity and structure of the free boundary. 2024. Preprint, arXiv:2410.23976v1], we show that each Λi is a nonpositive Radon measure, the negative of the measure produced by the subharmonicity of ui, and that Λi∈Wloc−1,p, 1<p<4. On the smooth two-phase part of the free boundary, it equals the negative jump of the one-sided normal derivative across the interface. Using the optimal Hölder regularity of minimizers, with exponent 3/4, we then prove that these measures assign no mass to the singular set of the free boundary. This upgrades the local jump identity to a global one: |Λi|≪Hd−1⌞Γ, with density on the regular part given by the normal-derivative jump.