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 𝒦 u=χ{u>0}, 𝒦=Δx+x·∇y−∂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 C𝒦1,β hypersurface for some β∈(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(r1+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 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·∇y−∂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.
Related
- Regularity of Free Boundary for Monge-Ampère Obstacle Problem up to the Boundary
- Wasserstein Stability and Free Boundaries in Measure-Parameterized Bilevel Obstacle Problems
- Mean curvature bounds for the obstacle problem
- A Schiffer-type inverse obstacle problem for the planar Schrödinger equation with a single measurement
- Global solutions to the fully nonlinear obstacle problem
BibTeX
@misc{nystrom2026higher,
title = {{Higher regularity of regular free boundaries for the Kolmogorov obstacle problem}},
author = {Kaj Nyström},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29676},
eprint = {2609.29676},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29676},
}
RIS
TY - UNPB
TI - Higher regularity of regular free boundaries for the Kolmogorov obstacle problem
AU - Kaj Nyström
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.29676
UR - https://arxiv.org/abs/2609.29676
AB - We establish higher regularity of the regular free boundary for non-negative solutions of the Kolmogorov obstacle problem 𝒦 u=χ_{u>0}, 𝒦=Δ_x+x·∇_y−∂_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_𝒦^1,β hypersurface for some β∈(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 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·∇_y−∂_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.
ER -