{
  "id": "1804.03537",
  "version": "v2",
  "published": "2018-04-10T13:50:34.000Z",
  "updated": "2018-10-29T23:22:57.000Z",
  "title": "Quantitative a Priori Estimates for Fast Diffusion Equations with Caffarelli-Kohn-Nirenberg weights. Harnack inequalities and Hölder continuity",
  "authors": [
    "Matteo Bonforte",
    "Nikita Simonov"
  ],
  "comment": "53 pages, 2 figures. This is a much improved version of the older manuscript, thanks to the suggestions of the anonymous referee",
  "categories": [
    "math.AP"
  ],
  "abstract": "We study a priori estimates for a class of non-negative local weak solution to the weighted fast diffusion equation $u_t = |x|^{\\gamma} \\nabla\\cdot (|x|^{-\\beta} \\nabla u^m)$, with $0 < m <1$ posed on cylinders of $(0,T)\\times{\\mathbb R}^N$. The weights $|x|^{\\gamma}$ and $|x|^{-\\beta}$, with $\\gamma < N$ and $\\gamma -2 < \\beta \\leq \\gamma(N-2)/N$ can be both degenerate and singular and need not belong to the class $\\mathcal{A}_2$, a typical assumption for this kind of problems. This range of parameters is optimal for the validity of a class of Caffarelli-Kohn-Nirenberg inequalities, which play the role of the standard Sobolev inequalities in this more complicated weighted setting. The weights that we consider are not translation invariant and this causes a number of extra difficulties and a variety of scenarios: for instance, the scaling properties of the equation change when considering the problem around the origin or far from it. We therefore prove quantitative - with computable constants - upper and lower estimates for local weak solutions, focussing our attention where a change of geometry appears. Such estimates fairly combine into forms of Harnack inequalities of forward, backward and elliptic type. As a consequence, we obtain H\\\"older continuity of the solutions, with a quantitative (even if non-optimal) exponent. Our results apply to a quite large variety of solutions and problems. The proof of the positivity estimates requires a new method and represents the main technical novelty of this paper. Our techniques are flexible and can be adapted to more general settings, for instance to a wider class of weights or to similar problems posed on Riemannian manifolds, possibly with unbounded curvature. In the linear case, $m=1$, we also prove quantitative estimates, recovering known results in some cases and extending such results to a wider class of weights.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2018-10-29T23:22:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B45",
      "35B65",
      "35K55",
      "35K67",
      "35K65"
    ],
    "keywords": [
      "fast diffusion equation",
      "priori estimates",
      "harnack inequalities",
      "hölder continuity",
      "caffarelli-kohn-nirenberg weights"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 53,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}