AI 中文总结
针对具L¹数据的双重奇异1-拉普拉斯问题，在极小假设下证明全局BV解存在性与比较原理，引入利用1-拉普拉斯算子刚性结构的新方法，可处理高度奇异项。
AI 中文摘要
本研究对如下问题开展全面分析：在有界Lipschitz区域Ω⊂ℝⁿ内，满足-Δ₁u + g(u)|Du| = h(u)f，边界∂Ω上u=0，其中f∈L¹(Ω)为非负数据，g、h为(0,∞)上非负连续函数，可能在原点处奇异。在g在原点附近可积、h在无穷远处有界的极小假设下，探讨全局BV(Ω)解的存在性；还在h满足适当单调性假设时证明了比较原理。该框架无需对h在原点附近施加增长限制，可容纳高度奇异项。为处理这些非线性项，引入了利用1-拉普拉斯算子刚性结构的新方法。
英文摘要
In this work, we conduct a comprehensive study of problem \begin{equation*} \begin{cases} -Δ_1 u + g(u)|Du| = h(u)f & \text{in }Ω, u=0 & \text{on } \partialΩ, \end{cases} \end{equation*} where $Ω\subset \mathbb{R}^N$ is a bounded Lipschitz domain, $f\in L^1(Ω)$ is a nonnegative datum, and $g,h$ are nonnegative continuous functions on $(0,\infty)$ that may be singular at the origin. Under the minimal assumptions that $g$ is integrable near zero and $h$ is bounded at infinity, we explore the existence of a global $BV(Ω)$ solution. Furthermore, a comparison principle is proved under suitable monotonicity assumptions on $h$. This framework avoids any growth restrictions on $h$ near the origin, thus allowing for highly singular terms. To handle these nonlinearities, we introduce a novel approach that takes advantage of the rigid structure of the 1-Laplacian operator.
Comments24 pages
