AI 中文总结
研究有界区域中各向异性极小曲面方程狄利克雷问题，通过建立先验梯度估计等方法，证明其唯一可解性、广义极小值的相关性质，还解释了变分形式产生诺伊曼边界条件及该问题无需边界曲率约束的原因。
AI 中文摘要
本文研究有界区域中各向异性极小曲面方程的狄利克雷问题。在非负边界各向异性平均曲率的自然假设下，我们建立了连续边界数据狄利克雷问题的唯一可解性。为此，建立了一个基本的先验梯度估计，这也使我们能够证明线性增长的整体解的弱伯恩斯坦定理。此外，使用变分法中的直接方法，我们证明了具有 \(L^1(\partial\Omega)\) 边界数据的 \(BV(\Omega)\) 中广义极小值的存在性和局部利普希茨正则性。我们还发现这种变分形式自然地产生了一个诺伊曼边界条件，从几何上解释了为什么诺伊曼问题不需要边界曲率约束。
英文摘要
This paper investigates the Dirichlet problem for the anisotropic minimal surface equation in a bounded domain. Under the natural assumption of non-negative boundary anisotropic mean curvature, we establish the unique solvability of the Dirichlet problem for continuous boundary data. To achieve this, an essential a priori gradient estimate is established, which also allows us to prove a weak version of Bernstein's theorem for entire solutions under a sharp, one-sided linear growth assumption. Moreover, using the direct method in the calculus of variations, we prove the existence and local Lipschitz regularity of generalized minimizers in $BV(Ω)$ with $L^1(\partialΩ)$ boundary data. We also find that this variational formulation naturally yields a Neumann-type boundary condition, geometrically explaining why the Neumann problem requires no boundary curvature constraints.
CommentsIn the proof of Theorem 1.1, we have added a more detailed estimate for the term ψ'F_{ξ_iξ_j}(ν)d_{ij}. Furthermore, the choice of the test function in Lemma 4.1 is modified, yielding a precise domain of integration
