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A parabolic free boundary problem with Bernoulli type condition on the free boundary
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Andersson, John and Weiss, Georg S. (2009) A parabolic free boundary problem with Bernoulli type condition on the free boundary. Journal für die reine und angewandte Mathematik (Crelles Journal), Vol.2009 (No.627). pp. 213-235. doi:10.1515/CRELLE.2009.016 ISSN 0075-4102.
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Official URL: http://dx.doi.org/10.1515/CRELLE.2009.016
Abstract
Consider the parabolic free boundary problem
Δu – ∂ t u = 0 in {u > 0}, |∇u| = 1 on ∂{u > 0}.
For a realistic class of solutions, containing for example all limits of the singular perturbation problem
Δuε – ∂ t uε = βε (uε ) as ε → 0,
we prove that one-sided flatness of the free boundary implies regularity.
In particular, we show that the topological free boundary ∂{u > 0} can be decomposed into an open regular set (relative to ∂{u > 0}) which is locally a surface with Hölder-continuous space normal, and a closed singular set.
Our result extends the main theorem in the paper by H. W. Alt-L. A. Caffarelli (1981) to more general solutions as well as the time-dependent case. Our proof uses methods developed in H. W. Alt-L. A. Caffarelli (1981), however we replace the core of that paper, which relies on non-positive mean curvature at singular points, by an argument based on scaling discrepancies, which promises to be applicable to more general free boundary or free discontinuity problems.
Item Type: | Journal Article | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||
Journal or Publication Title: | Journal für die reine und angewandte Mathematik (Crelles Journal) | ||||
Publisher: | Walter de Gruyter GmbH & Co. KG | ||||
ISSN: | 0075-4102 | ||||
Official Date: | February 2009 | ||||
Dates: |
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Volume: | Vol.2009 | ||||
Number: | No.627 | ||||
Page Range: | pp. 213-235 | ||||
DOI: | 10.1515/CRELLE.2009.016 | ||||
Status: | Peer Reviewed | ||||
Publication Status: | Published | ||||
Access rights to Published version: | Restricted or Subscription Access |
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