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Creators/Authors contains: "Teixeira, Eduardo"

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  1. ; (, Transactions of the American Mathematical Society)
    We consider a generalization of the Bernoulli free boundary problem where the underlying differential operator is a nonlocal, non-translation-invariant elliptic operator of order 2 s ∈<#comment/> ( 0 , 2 ) 2s\in (0,2) . Because of the lack of translation invariance, the Caffarelli-Silvestre extension is unavailable, and we must work with the nonlocal problem directly instead of transforming to a thin free boundary problem. We prove global Hölder continuity of minimizers for both the one- and two-phase problems. Next, for the one-phase problem, we show Hölder continuity at the free boundary with the optimal exponent s s . We also prove matching nondegeneracy estimates. A key novelty of our work is that all our findings hold without requiring any regularity assumptions on the kernel of the nonlocal operator. This characteristic makes them crucial in the development of a universal regularity theory for nonlocal free boundary problems. 
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    Free, publicly-accessible full text available June 1, 2026
  2. ; (, Calculus of Variations and Partial Differential Equations)
    Free, publicly-accessible full text available December 1, 2025
  3. ; (, SIAM Journal on Mathematical Analysis)
    We prove the existence of an open set minimizing the first Dirichlet eigenvalue of an elliptic operator with bounded, measurable coefficients, over all open sets of a given measure. Our proof is based on a free boundary approach: we characterize the eigenfunction on the optimal set as the minimizer of a penalized functional, and derive openness of the optimal set as a consequence of a Hölder estimate for the eigenfunction. We also prove that the optimal eigenfunction grows at most linearly from the free boundary, i.e., it is Lipschitz continuous at free boundary points. 
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