Kolmogorov障碍问题正则自由边界的更高正则性
Higher regularity of regular free boundaries for the Kolmogorov obstacle problem
- Uppsala University（乌普萨拉大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
针对Kolmogorov障碍问题，在接触集满足定量厚度条件下，证明了正则自由边界为非特征C^{1,β}超曲面，通过调和替换与边界衰减估计实现平坦性改进。
AI中文摘要：
我们建立了Kolmogorov障碍问题\\[ \mathcal K u=\chi_{\{u>0\}}, \qquad \mathcal K=\Delta_x+x\cdot\nabla_y-\partial_t \\]的非负解的正则自由边界的更高正则性。假设接触集在自由边界点处满足定量厚度条件。我们证明，在该点的邻域内，自由边界是某个\beta\in(0,1)的非特征\\(C_{\mathcal K}^{1,\beta}\\)超曲面。等价地，其向内非退化法向量关于内在Kolmogorov距离是Hölder连续的，并且图形与其切超平面的偏差在内在尺度r下为\\(O(r^{1+\beta})\\)阶。证明结合了半空间爆破和正则点处的可微性理论与渐近圆柱形内在Lipschitz域中的边界Harnack估计。一个主要困难是确定法向量的扩散导数不满足齐次Kolmogorov方程。我们引入了\\(\mathcal K\\)-调和替换论证，该论证将交换子源转化为内在爆破下的\\(O(r)\\)误差，并产生导数商振荡的收缩估计。这给出了非退化法向量的Hölder连续性。此外，输运导数\\(Yu=(x\cdot\nabla_y-\partial_t)u\\)的边界衰减估计提供了耦合时间-输运方向所需的改进。将这些估计与三次变量中可用的控制相结合，产生了完全的平坦性内在改进。
英文摘要：
We establish higher regularity of the regular free boundary for non-negative solutions of the Kolmogorov obstacle problem \[ \mathcal K u=χ_{\{u>0\}}, \qquad \mathcal K=Δ_x+x\cdot\nabla_y-\partial_t. \] Assume that the contact set satisfies a quantitative thickness condition at a free-boundary point. We prove that, in a neighborhood of that point, the free boundary is a non-characteristic \(C_{\mathcal K}^{1,β}\) hypersurface for some \(β\in(0,1)\). Equivalently, its inward non-degenerate normal is Hölder continuous with respect to the intrinsic Kolmogorov distance, and the deviation of the graph from its tangent hyperplane is of order \(O(r^{1+β})\) at intrinsic scale \(r\). The proof combines the half-space blow-up and differentiability theory at regular points with boundary Harnack estimates in asymptotically cylindrical intrinsic Lipschitz domains. A principal difficulty is that the diffusion derivatives determining the normal do not satisfy the homogeneous Kolmogorov equation. We introduce a \(\mathcal K\)-harmonic-replacement argument that converts the commutator sources into an \(O(r)\) error under intrinsic blow-up and yields a contraction estimate for the oscillations of derivative quotients. This gives Hölder continuity of the non-degenerate normal. In addition, a boundary decay estimate for the transport derivative $Yu=(x\cdot\nabla_y-\partial_t)u$ provides the required improvement in the coupled time-transport direction. Combining these estimates with the available control in the degree-three variables yields a full intrinsic improvement of flatness.
