Generalized Carleson perturbations of elliptic operators and applications
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- by Joseph Feneuil and Bruno Poggi PDF
- Trans. Amer. Math. Soc. 375 (2022), 7553-7599 Request permission
Abstract:
We extend in two directions the notion of perturbations of Carleson type for the Dirichlet problem associated to an elliptic real second-order divergence-form (possibly degenerate, not necessarily symmetric) elliptic operator. First, in addition to the classical perturbations of Carleson type, that we call additive Carleson perturbations, we introduce scalar-multiplicative and antisymmetric Carleson perturbations, which both allow non-trivial differences at the boundary. Second, we consider domains which admit an elliptic PDE in a broad sense: we count as examples the 1-sided NTA (a.k.a. uniform) domains satisfying the capacity density condition, the 1-sided chord-arc domains, the domains with low-dimensional Ahlfors-David regular boundaries, and certain domains with mixed-dimensional boundaries; thus our methods provide a unified perspective on the Carleson perturbation theory of elliptic operators.
Our proofs do not introduce sawtooth domains or the extrapolation method. We also present several applications to some Dahlberg-Kenig-Pipher operators, free-boundary problems, and we provide a new characterization of $A_{\infty }$ among elliptic measures.
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Additional Information
- Joseph Feneuil
- Affiliation: Laboratoire de mathématiques d’Orsay, Université Paris-Saclay, CNRS, 91405 Orsay, France
- MR Author ID: 1119999
- ORCID: 0000-0001-5505-4450
- Email: joseph.feneuil@gmail.com
- Bruno Poggi
- Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
- MR Author ID: 1339605
- Email: bgpoggi.math@gmail.com
- Received by editor(s): June 7, 2021
- Published electronically: August 24, 2022
- Additional Notes: J.F. was partially supported by the Simons Foundation grant 601941, GD. B.P. was supported by the University of Minnesota Doctoral Dissertation Fellowship grant and in part by the Simons Collaborations in MPS 563916, SM. The authors would like to thank Svitlana Mayboroda and Max Engelstein for insightful conversations
- © Copyright 2022 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 375 (2022), 7553-7599
- MSC (2000): Primary 31B25, 35J25, 35J70
- DOI: https://doi.org/10.1090/tran/8635