发表机构
Kumoh National Institute of Technology（龟尾国立技术大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究具有$L^2$-梯度扰动和可积零阶系数的线性椭圆方程，证明在粗糙假设下弱解的适定性及定量能量和$L^2$估计仍成立，并刻画源项与系数的可积性权衡。
AI 中文摘要
本文研究如下形式的线性椭圆Dirichlet问题弱解的存在性、唯一性和定量估计：\\[-\operatorname{div}(\gamma \nabla u)+\langle \nabla\phi+\mathbf{H},\nabla u\rangle+(c+\alpha)u=f \quad\text{在}U\text{中}, \quad u=0 \quad\text{在}\partial U\text{上},\\]其中$U\subset \mathbb{R}^d$有界，$\gamma\in[1,\infty)$为常数，$\phi\in H^{1,2}(U)\cap L^\infty(U)$，$\mathbf{H}\in L^p(U,\mathbb{R}^d)$对某个$p \in (d, \infty)$，且$c\in L^1(U)$满足$c\ge0$。该设置的一个关键特征是漂移项包含低正则项$\nabla\phi$，仅假设其属于$L^2(U,\mathbb{R}^d)$，而零阶系数仅为可积。我们证明，即使在这些粗糙假设下，适定性和定量能量及$L^2$估计仍然成立。此外，通过使用先前建立的插值结果，我们刻画了源项$f$的可积性与零阶系数$c$的可积性之间的权衡，并表明在这些插值假设下，适定性及相应的定量估计成立。
英文摘要
In this paper, we study the existence, uniqueness, and quantitative estimates for weak solutions to linear elliptic Dirichlet problems of the form \[ -\operatorname{div}(γ\nabla u)+\langle \nablaϕ+\mathbf{H},\nabla u\rangle+(c+α)u=f \quad\text{ in }U, \quad\; u=0 \quad\text{on }\partial U, \] where $U\subset \mathbb{R}^d$ is bounded, $γ\in[1,\infty)$ is a constant, $ϕ\in H^{1,2}(U)\cap L^\infty(U)$, $\mathbf{H}\in L^p(U,\mathbb{R}^d)$ for some $p \in (d, \infty)$, and $c\in L^1(U)$ with $c\ge0$. A key feature of this setting is that the drift contains the low-regularity term $\nablaϕ$, which is only assumed to belong to $L^2(U,\mathbb{R}^d)$, while the zero-order coefficient is merely integrable. We prove that, even under these rough assumptions, well-posedness and quantitative energy and $L^2$ estimates remain valid. In addition, by using a previously established interpolation result, we characterize a trade-off between the integrability of the source term $f$ and that of the zero-order coefficient $c$, and show that well-posedness together with the corresponding quantitative estimates hold under these interpolated assumptions.
Comments11 pages
