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We study the existence and structure of branch points in two-phase freeboundary problems. More precisely, we construct a family of minimizers to anAlt- Caffarelli-Friedman type functional whose free boundaries contain branchpoints in the strict interior of the domain. We also give an example showingthat branch points in the free boundary of almost-minimizers of the samefunctional can have very little structure. This last example stands in contrastwith recent results of De Philippis- Spolaor-Velichkov on the structure ofbranch points in the free boundary of stationary solutions.

Branch Points for (Almost-)Minimizers of Two-Phase Free Boundary Problems