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September 10, 2025Open Access

An alternative proof of the solvability of the Lᵖ-regularity problem for Dahlberg-Kenig-Pipher operators on {R}ⁿ_+

Key Points

The result addresses solvability for the l^p regularity problem in uniformly elliptic operators.
Incorporating Carleson measure conditions, we present a simpler proof of the solvability of the Dirichlet problem.
The regularity problem is also expanded to include weighted uniformly elliptic operators on R^n excluding R^d.
A significant result is proven for d less than n-1, expanding the applicability of the regularity theory.

Abstract

Abstract In this article, we present a simpler and alternative proof of the solvability of the regularity problem—that is, the Dirichlet problem with boundary data in Ẇ^1, p W ˙ 1, p —for uniformly elliptic operators on {R}ⁿ_+ R + n under a (possibly large) Carleson measure condition. In addition, we slightly expand the class of operators for which the regularity problem is solvable, and establish an analogous result for weighted uniformly elliptic operators on {R}ⁿ {R}ᵈ R n \ R d, where d d n - 1.

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Joseph Feneuil (Fri,) studied this question.

synapsesocial.com/papers/68c1d98554b1d3bfb60fb47e https://doi.org/https://doi.org/10.1007/s00209-025-03837-0

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