Dirichlet problem in domains with lower dimensional boundaries

2018/10/16 by Joseph Feneuil, Svitlana Mayboroda, Feneuil, Joseph +3 · 1 citation
Mathematics · #35J25 #35J70 #Advanced Harmonic Analysis Research #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.1810.06805

openalex publication_date 2018/10/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

The present paper pioneers the study of the Dirichlet problem with L^q boundary data for second order operators with complex coefficients in domains with lower dimensional boundaries, e.g., in Ω:= \mathbb Rⁿ ∖ \mathbb R^d with d1 provided that the coefficients satisfy the small Carleson norm condition. Even in the context of the classical case d=n-1, (the analogues of) our results are new. The conditions on the coefficients are more relaxed than the previously known ones (most notably, we do not impose any restrictions whatsoever on the first n-1 rows of the matrix of coefficients) and the results are more general. We establish local rather than global estimates between the square function and the non-tangential maximal function and, perhaps even more importantly, we establish new Moser-type estimates at the boundary and improve the interior ones.

Citations

• The L p Dirichlet problem for second order elliptic operators and a p-adapted square function
• A New Approach to Absolute Continuity of Elliptic Measure, with Applications to Non-symmetric Equations

Cited by

• The Poisson-Dirichlet problem in domains with Ahlfors regular boundary
• The L^p Dirichlet boundary problem for second order Elliptic Systems with rough coefficients

Related

• Carleson perturbations of elliptic operators on domains with low dimensional boundaries
• Carleson estimates for the Green function on domains with lower dimensional boundaries
• The Boundary value problems for second order elliptic operators satisfying a Carleson condition
• Boundary Value Problems for Second‐Order Elliptic Operators Satisfying a Carleson Condition
• Regularity and Neumann problems for operators with real coefficients satisfying Carleson condition