The Dirichlet Problem for elliptic equations with singular drift terms
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909979391295488 |
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| author | Hofmann, Steve |
| author_facet | Hofmann, Steve |
| contents | We establish $L^p$ solvability of the Dirichlet problem, for some finite $p$, in a 1-sided chord-arc domain $Ω$ (i.e., a uniform domain with Ahlfors-David regular boundary), for elliptic equations of the form \[ Lu=-\text{div}(A\nabla u) + {\bf B}\cdot \nabla u=:L_0 u+ {\bf B}\cdot \nabla u=0, \] 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 $|{\bf B}(X)|\lesssim \text{dist}(X,\partial Ω)^{-1}$, and that $|{\bf B}(X)|^2\text{dist}(X,\partial Ω) dX$ is a Carleson measure in $Ω$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_03665 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Dirichlet Problem for elliptic equations with singular drift terms Hofmann, Steve Analysis of PDEs 35J15, 35J25, 42B37 We establish $L^p$ solvability of the Dirichlet problem, for some finite $p$, in a 1-sided chord-arc domain $Ω$ (i.e., a uniform domain with Ahlfors-David regular boundary), for elliptic equations of the form \[ Lu=-\text{div}(A\nabla u) + {\bf B}\cdot \nabla u=:L_0 u+ {\bf B}\cdot \nabla u=0, \] 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 $|{\bf B}(X)|\lesssim \text{dist}(X,\partial Ω)^{-1}$, and that $|{\bf B}(X)|^2\text{dist}(X,\partial Ω) dX$ is a Carleson measure in $Ω$. |
| title | The Dirichlet Problem for elliptic equations with singular drift terms |
| topic | Analysis of PDEs 35J15, 35J25, 42B37 |
| url | https://arxiv.org/abs/2502.03665 |