Higher regularity of regular free boundaries for the Kolmogorov obstacle problem

Abstract

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.