实与复退化黑塞方程的刘维尔刚性
Liouville Rigidity for Real and Complex Degenerate Hessian Equations
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中文总结 AI 辅助
研究实与复退化黑塞方程的刘维尔刚性，通过刘维尔容许性这一递归几何条件，证明当且仅当容许集满足该条件时，相关方程的有界全局\(C^{0,\alpha}\)整体粘性解为常数，还给出了相关例子及各向异性构造。
中文摘要 AI 辅助
我们在粘性意义下证明了平移不变实和复黑塞方程的刘维尔刚性定理，其中偏微分方程由一个容许集\(\mathcal{A}\)编码。主要结构概念是刘维尔容许性，这是一个递归几何条件，要求每个商集要么是边界兼容的，要么属于一个终端类。我们的主要定理表明，当且仅当\(\mathcal{A}\)是刘维尔容许的时，\(\mathrm{Hess}_{\mathbb F}u\in\partial\mathcal{A}\)的每个有界、全局\(C^{0,\alpha}\)整体粘性解是常数；因此刘维尔型性质被刻画为容许集的几何性质。一类核心例子来自满足单调根序列条件的单变量加尔丁多项式的极化，产生混合初等对称容许集，并将标准\(k -\)黑塞方程作为单项式情形恢复。该框架还允许各向异性构造，包括线性拉回和容许集的交集。
英文摘要
We prove Liouville rigidity theorems for translation-invariant real and complex Hessian equations in the viscosity sense, where the PDE is encoded by an admissible set $\mathcal{A}$. The main structural notion is Liouville admissibility, a recursive geometric condition requiring each quotient set to be either boundary compatible or to fall into a terminal class. Our main theorem states that every bounded, globally $C^{0,α}$ entire viscosity solution of \[ \mathrm{Hess}_{\mathbb F}u\in\partial\mathcal{A} \] is constant if and only if $\mathcal{A}$ is Liouville admissible; thus the Liouville-type property is characterized as a geometric property of the admissible set. A central class of examples arises from polarizations of univariate Gårding polynomials satisfying the monotone root sequence condition, producing mixed elementary-symmetric admissible sets and recovering the standard $k$-Hessian equations as monomial cases. The framework also allows anisotropic constructions, including linear pullbacks and intersections of admissible sets.
