AI 中文总结
本文研究带哈密顿项的奇异与退化完全非线性椭圆方程粘性解的相关性质，建立其整体正则性、Harnack不等式、Liouville刚性及主特征值的统一理论。
AI 中文摘要
我们研究有界区域Ω中奇异或退化完全非线性椭圆方程Φ(x,|∇u|)F(D²u)-H(x,∇u)+c(x)|u|^{i(Φ)}u=h(x)的粘性解的正则性、Harnack不等式、Liouville刚性及主特征值，其中Φ描述对梯度的奇异或退化依赖，H为哈密顿项。首先，我们证明不含低阶项的Dirichlet问题的整体C^{1,γ}正则性，该论证结合了来自Alexandroff–Bakelman–Pucci不等式的整体L^∞估计、给出整体Lipschitz界的边界障碍，以及基于紧性的迭代近似格式。接着，我们通过从下方滑动形如-|x|^{1/2}的尖点函数，建立非负粘性解的加性Harnack不等式。在额外的齐次性假设下，这给出了ℝⁿ中的经典Harnack不等式和Liouville型定理。最后，在适当的齐次性与比较假设下，我们为完整算子发展了广义Dirichlet主特征值理论，证明了主特征函数的存在性，并通过极大值与极小值原理刻画相关特征值。这些结果为一大类带哈密顿项的奇异与退化完全非线性方程提供了关于整体正则性、Harnack估计、Liouville刚性及主特征值的统一框架。
英文摘要
We study regularity, Harnack inequalities, Liouville rigidity, and principal eigenvalues for viscosity solutions of singular or degenerate fully nonlinear elliptic equations $Φ(x,|\nabla u|)F(D^2u)-H(x,\nabla u)+c(x)|u|^{i(Φ)}u=h(x)$ in a bounded domain $Ω$, where $Φ$ describes the singular or degenerate dependence on the gradient and $H$ is a Hamiltonian. We first prove global $C^{1,γ}$ regularity for the Dirichlet problem without lower-order terms. The argument combines a global $L^\infty$ estimate from the Alexandroff--Bakelman--Pucci inequality, boundary barriers yielding a global Lipschitz bound, and a compactness-based iterative approximation scheme. We next establish an additive Harnack inequality for nonnegative viscosity solutions by sliding from below a cusp function of the form $-|x|^{1/2}$. Under an additional homogeneity assumption, this yields the classical Harnack inequality and Liouville-type theorems in $\mathbb{R}^n$. Finally, under suitable homogeneity and comparison assumptions, we develop a generalized Dirichlet principal-eigenvalue theory for the full operator. We prove the existence of principal eigenfunctions and characterize the associated eigenvalues through maximum and minimum principles. These results provide a unified framework for global regularity, Harnack estimates, Liouville rigidity, and principal eigenvalues for a broad class of singular and degenerate fully nonlinear equations with Hamiltonian terms.
