The Dirichlet problem for elliptic equations with singular drift terms
Abstract
We establish \(L^p\) solvability of the Dirichlet problem, for some finite \(p\), in a 1-sided chord-arc domain \(\Omega\) (i.e., a uniform domain with Ahlfors–David regular boundary), for elliptic equations of the form \[Lu=-\operatorname{div}(A\nabla u) +\mathbf{B}\cdot \nabla u=:L_0 u+ \mathbf{P}\cdot \nabla u=0,\] given that the analogous result holds (typically with a different value of \(p\)) for the homogeneous second order operator \(L_0\). Essentially, we assume that \(|\mathbf{B}(X)|\lesssim \operatorname{dist}(X,\partial\Omega)^{-1}\), and that \[|\mathbf{B}(X)|^2\operatorname{dist}(X,\partial\Omega)\, dX\] is a Carleson measure in \(\Omega\).
Fields of Study & Subjects
Advanced Mathematical Modeling in Engineering
Differential Equations and Boundary Problems
Differential Equations and Numerical Methods
Publication details
- Journal / source
- Annales Fennici Mathematici
- Volume
- 51
- Issue
- 2
- Pages
- 635–684
- Publication date
- 15 September 2026
- Language
- EN