{
  "id": "1005.3882",
  "version": "v1",
  "published": "2010-05-21T04:13:23.000Z",
  "updated": "2010-05-21T04:13:23.000Z",
  "title": "On the singularities of a free boundary through Fourier expansion",
  "authors": [
    "John Andersson",
    "Henrik Shahgholian",
    "Georg S. Weiss"
  ],
  "comment": "39 pages, 5 figures",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper we are concerned with singular points of solutions to the {\\it unstable} free boundary problem $$ \\Delta u = - \\chi_{\\{u>0\\}} \\qquad \\hbox{in} B_1. $$ The problem arises in applications such as solid combustion, composite membranes, climatology and fluid dynamics. It is known that solutions to the above problem may exhibit singularities - that is points at which the second derivatives of the solution are unbounded - as well as degenerate points. This causes breakdown of by-now classical techniques. Here we introduce new ideas based on Fourier expansion of the nonlinearity $\\chi_{\\{u>0\\}} $. The method turns out to have enough momentum to accomplish a complete description of the structure of the singular set in ${\\mathbb R}^3$. A surprising fact in ${\\mathbb R}^3$ is that although $$\\frac{u(r\\x)}{\\sup_{B_1}|u(r\\x)|}$$ can converge at singularities to each of the harmonic polynomials $$ xy, {x^2+y^2\\over 2}-z^2 \\textrm{and} z^2-{x^2+y^2\\over 2},$$ it may {\\em not} converge to any of the non-axially-symmetric harmonic polynomials $\\alpha((1+ \\delta)x^2 +(1- \\delta)y^2 - 2z^2)$ with $\\delta\\ne 1/2$. We also prove the existence of stable singularities in ${\\mathbb R}^3$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-05-21T04:13:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35R35"
    ],
    "keywords": [
      "fourier expansion",
      "singularities",
      "non-axially-symmetric harmonic polynomials",
      "free boundary problem",
      "problem arises"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 39,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1005.3882A"
    }
  }
}