{
  "abstract": "We give a self-contained account of the classical obstacle problem, written for graduate students who wish to enter the field of free boundary problems. Starting from a one-dimensional model and an explicit radially symmetric example, we set up the variational problem, show that minimizers exist and are unique, and prove interior $W^{2,p}$-regularity as well as the optimal $C^{1,1}$-regularity. The second half of the notes is devoted to the free boundary: we prove that it is porous, introduce blow-up limits, and use the Alt-Caffarelli-Friedman monotonicity formula to show that every free boundary point is either regular or singular. These notes grew out of a mini-course at the Rio Summer School on PDEs in 2020, and they are meant to be read alongside, not instead of, the monograph by Petrosyan, Shahgholian, and Uraltseva.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.37943",
  "author_total": 1,
  "authors": [
    {
      "name": "Andreas Minne"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e22bm9ch8jzrb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.37943",
  "fulltext_url": null,
  "id": "e22bm9ch8jzrb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.37943",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2610.37943",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e22bm9ch8jzrb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.37943",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.37943"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "An Introduction to the Obstacle Problem",
  "title_text": "An Introduction to the Obstacle Problem",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/e22bm9ch8jzrb",
  "venue": "arXiv",
  "versions": []
}