AI 中文总结
该研究针对凸有界区域上带齐次毛细边界条件的给定洛伦兹平均曲率问题，证明其存在唯一$W^{2,2}$正则弱解，且该解为相关泛函的唯一极大值点，关键在于证明解不存在光线段。
AI 中文摘要
我们研究在$\boldsymbol{\text{R}}^m$的凸有界区域$\boldsymbol{\text{\textit{Ω}}}$上，带有有界右端项和齐次毛细边界条件的给定洛伦兹平均曲率问题。我们证明该问题存在唯一的$W^{2,2}$正则弱解$\boldsymbol{\textit{u}}$，其均值为零，且对某个仅依赖于数据的$\theta \boldsymbol{\text{∈}}(0,1)$，满足$|\boldsymbol{\textit{Du}}| \boldsymbol{\text{≤}} 1 - \theta$。该$\boldsymbol{\textit{u}}$也是相关泛函的唯一极大值点。证明该极大值点为弱解的关键步骤在于，证明其不存在光线段，即满足$|\boldsymbol{\textit{Du}}| \boldsymbol{\text{=}} 1$的线段，这一结论对任意有界毛细边界数据均成立，因此本身也是一个值得关注的结果。
英文摘要
We address the prescribed Lorentzian mean curvature problem over a convex bounded domain $Ω$ of $\mathbb R^m$ with bounded right-hand side and homogeneous capillary boundary condition. We prove that the problem has a unique $W^{2,2}$-regular weak solution $u$ with zero mean and that $|Du| \leq 1 - θ$ for some $θ\in(0,1)$ only depending on the data. Such $u$ is also the unique maximizer of an associated functional. A key step in proving that the maximizer is a weak solution consists in showing that it has no light segments, i.e. segments along which $|Du| = 1$. This holds for arbitrary bounded capillary boundary data and can thus be an interesting result on its own.
