Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Regularity and nondegeneracy for nonlocal Bernoulli problems with variable kernels
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by Stanley Snelson and Eduardo V. Teixeira;
Trans. Amer. Math. Soc. 378 (2025), 4109-4127
DOI: https://doi.org/10.1090/tran/9343
Published electronically: March 19, 2025

Abstract:

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 $2s\in (0,2)$. 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 $s$. 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.
References
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Bibliographic Information
  • Stanley Snelson
  • Affiliation: Department of Mathematics and Systems Engineering, 150 W. University Blvd, Florida Institute of Technology, Melbourne, Florida 32901
  • MR Author ID: 709090
  • ORCID: 0000-0002-6830-1606
  • Email: ssnelson@fit.edu
  • Eduardo V. Teixeira
  • Affiliation: Department of Mathematics, University of Central Florida, 4393 Andromeda Loop N, Orlando, Florida 32816
  • MR Author ID: 710372
  • ORCID: 0000-0003-1583-3243
  • Email: eduardo.teixeira@ucf.edu
  • Received by editor(s): April 22, 2024
  • Received by editor(s) in revised form: August 25, 2024
  • Published electronically: March 19, 2025
  • Additional Notes: The first author was partially supported by NSF grant DMS-2213407 and a Collaboration Grant from the Simons Foundation, Award #855061.
  • © Copyright 2025 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 378 (2025), 4109-4127
  • MSC (2020): Primary 35R35; Secondary 35R09
  • DOI: https://doi.org/10.1090/tran/9343
  • MathSciNet review: 4907951