{
  "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.29676",
  "author_total": 1,
  "authors": [
    {
      "name": "Kaj Nyström"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/czz9qwg9a8agt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.29676",
  "fulltext_url": null,
  "id": "czz9qwg9a8agt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.29676",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "openalex",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7214329232",
  "pdf_url": "https://arxiv.org/pdf/2609.29676",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/czz9qwg9a8agt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.29676",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.29676"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Higher regularity of regular free boundaries for the Kolmogorov obstacle problem",
  "title_text": "Higher regularity of regular free boundaries for the Kolmogorov obstacle problem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/czz9qwg9a8agt",
  "venue": "arXiv",
  "versions": []
}