AI 中文总结
本文针对Kolmogorov方程，在可依赖高阶变量的渐近圆柱Lipschitz区域建立边界Harnack不等式，突破早期理论的对称性限制，通过多类估计与论证排除异常构型，得到正解的相关性质。
AI 中文摘要
我们在非特征内禀Lipschitz图区域中为Kolmogorov方程\\(\mathcal Ku=\Delta_xu+x\cdot\nabla_yu-\partial_tu=0\\)的非负解建立边界Harnack不等式，该区域的定义函数可依赖于高阶变量\\(y_m\\)。早期理论要求该函数与\\(y_m\\)无关，我们用定量渐近圆柱性替代这种对称性，其由尺度不变的\\(C^{0,1/3}_{y_m}\\)缺陷度量。只要该缺陷足够小，扰动比较定理就适用，因此对于满足\\(\alpha>1/3\\)的\\(C^{0,\alpha}_{y_m}\\)图，以及在消失的临界圆柱模量下，该定理在所有足够小的尺度上均成立。我们得到正解商的局部可比性和内禀Hölder连续性，以及其对数商的Hölder估计。证明结合了不依赖\\(y_m\\)的局部边界估计，以及圆柱爆破极限的全局刚性论证。核心问题是正向与反向参考值的均匀平衡：若该平衡在坍缩尺度下失效，最大尺度选择与第二次重标度会在无界圆柱区域中产生多项式增长的非负Dirichlet解，该解在反向参考点处消失但在正向参考点处不消失。我们利用边界能量与均值估计、多项式增长解空间的有限维性，以及\\(y_m\\)的平移不变性排除该构型。所得有限维平移表示产生对\\(y_m\\)的实解析依赖性，零的传播、沿不变纤维的解析延拓以及Tikhonov唯一性论证随后迫使极限解恒为零。指数\\(1/3\\)是临界的，因为\\(y_m\\)具有三次齐次次数。
英文摘要
We establish boundary Harnack inequalities for non-negative solutions of the Kolmogorov equation \[ \mathcal Ku=Δ_xu+x\cdot\nabla_yu-\partial_tu=0 \] in non-characteristic intrinsic Lipschitz graph domains whose defining functions may depend on the higher-order variable \(y_m\). Earlier theory required independence of \(y_m\). We replace this symmetry by quantitative asymptotic cylindricality, measured by a scale-invariant \(C^{0,1/3}_{y_m}\) defect. The perturbative comparison theorem applies whenever this defect is sufficiently small, and hence at all sufficiently small scales for \(C^{0,α}_{y_m}\) graphs with \(α>1/3\), as well as under a vanishing critical cylindrical modulus. We obtain local comparability and intrinsic Hölder continuity of quotients of positive solutions, together with a Hölder estimate for their logarithmic quotient. The proof combines local boundary estimates, valid without \(y_m\)-independence, with a global rigidity argument for cylindrical blow-up limits. The central issue is uniform balance of forward and backward reference values. If this balance fails at collapsing scales, maximal-scale selection and a second rescaling produce a non-negative polynomial-growth Dirichlet solution in an unbounded cylindrical domain that vanishes at the backward reference point but not at the forward one. We exclude this configuration using boundary-energy and mean-value estimates, finite-dimensionality of spaces of polynomial-growth solutions, and translation invariance in \(y_m\). The resulting finite-dimensional translation representation yields real-analytic dependence on \(y_m\). Propagation of zeros, analytic continuation along invariant fibres, and a Tikhonov uniqueness argument then force the limiting solution to vanish identically. The exponent \(1/3\) is critical because \(y_m\) has homogeneous degree three.
