Dirichlet and Neumann problems for elliptic equations with singular drifts on Lipschitz domains
- 주제(키워드) elliptic equations , singular drift terms , Dirichlet boundary problem , Neumann boundary problem , Lipschitz domain
- 발행기관 서강대학교 일반대학원
- 지도교수 김현석
- 발행년도 2019
- 학위수여년월 2019. 2
- 학위명 석사
- 학과 및 전공 일반대학원 수학과
- 실제URI http://www.dcollection.net/handler/sogang/000000063756
- UCI I804:11029-000000063756
- 본문언어 한국어
- 저작권 서강대학교 논문은 저작권보호를 받습니다.
초록/요약
In this thesis, we consider the Dirichlet and Neumann problems for second-order linear elliptic equations:\[ -\triangle u +\Div(u\boldb) =f \quad\text{ and }\quad -\triangle v -\boldb \cdot \nabla v =g \] in a bounded Lipschitz domain $\Omega$ in $\mathbb{R}^n$ $(n\geq 3)$, where $\boldb:\Omega \rightarrow \mathbb{R}^n$ is a given vector field. Under the assumption that $\boldb \in L^{n}(\Omega)^n$, we first establish existence and uniqueness of solutions in $L_{\alpha}^{p}(\Omega)$ for the Dirichlet and Neumann problems. Here $L_{\alpha}^{p}(\Omega)$ denotes the Sobolev space (or Bessel potential space) with the pair $(\alpha,p)$ satisfying certain conditions. These results extend the classical works of Jerison-Kenig \cite{MR1331981} and Fabes-Mendez-Mitrea \cite{MR1658089} for the Poisson equation. We also prove existence and uniqueness of solutions of the Dirichlet problem with boundary data in $L^{2}(\partial\Omega)$.
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