JournalsifbVol. 15, No. 4pp. 477–499

Optimal regularity for the parabolic no-sign obstacle type problem

  • John Andersson

    KTH Royal Institute of Technology, Stockholm, Sweden
  • Erik Lindgren

    KTH Royal Institute of Technology, Stockholm, Sweden
  • Henrik Shahgholian

    KTH Royal Institute of Technology, Stockholm, Sweden
Optimal regularity for the parabolic no-sign obstacle type problem cover
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Abstract

We study the parabolic free boundary problem of obstacle type

Δuut=fχ{u0}.\Delta u-\frac{\partial u}{\partial t}= f\chi_{\{u\ne 0\}}.

Under the condition that f=Hvf=Hv for some function vv with bounded second order spatial derivatives and bounded first order time derivative, we establish the same regularity for the solution uu. Both the regularity and the assumptions are optimal. Using this result and assuming that ff is Dini continuous, we prove that the free boundary is, near so called low energy points, a C1C^1 graph. Our result completes the theory for this type of problems for the heat operator.

Cite this article

John Andersson, Erik Lindgren, Henrik Shahgholian, Optimal regularity for the parabolic no-sign obstacle type problem. Interfaces Free Bound. 15 (2013), no. 4, pp. 477–499

DOI 10.4171/IFB/311