We study the existence and structure of branch points in two-phase free boundary problems. More precisely, we construct a family of minimizers to an Alt–Caffarelli–Friedman-type functional whose free boundaries contain branch points in the strict interior of the domain. We also give an example showing that branch points in the free boundary of almost-minimizers of the same functional can have very little structure. This last example stands in contrast with recent results of De Philippis, Spolaor and Velichkov on the structure of branch points in the free boundary of stationary solutions.
Bibliographical noteFunding Information:
G.D. was partially supported by the ANR, programme blanc GEOMETRYA, ANR-12-BS01-0014, the European H2020 Grant GHAIA 777822, and the Simons Collaborations in MPS Grant 601941, GD. M.E. has been partially supported by the NSF grant DMS 2000288. M.S.V.G. has been partially supported by the NSF grant DMS 2054282. T.T. was partially supported by the Craig McKibben & Sarah Merner Professor in Mathematics, and by NSF grant DMS-1954545.
© The Author(s), 2023.