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

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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
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of 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:	Date Event February 2009 Published
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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