Andersson, John, Shahgholian, Henrik and Weiss, Georg S.. (2010) Uniform regularity close to cross singularities in an unstable free boundary problem. Communications in Mathematical Physics, Vol.296 (No.1). pp. 251-270. ISSN 0010-3616

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Abstract

We introduce a new method for the analysis of singularities in the unstable problem Delta u = chi{u> 0}, which arises in solid combustion as well as in the composite membrane problem. Our study is confined to points of "supercharacteristic" growth of the solution, i.e. points at which the solution grows faster than the characteristic/invariant scaling of the equation would suggest. At such points the classical theory is doomed to fail, due to incompatibility of the invariant scaling of the equation and the scaling of the solution. In the case of two dimensions our result shows that in a neighborhood of the set at which the second derivatives of u are unbounded, the level set {u = 0} consists of two C-1-curves meeting at right angles. It is important that our result is not confined to the minimal solution of the equation but holds for all solutions.

Item Type:	Journal Article
Subjects:	Q Science > QC Physics
Divisions:	Faculty of Science > Mathematics
Journal or Publication Title:	Communications in Mathematical Physics
Publisher:	Springer
ISSN:	0010-3616
Date:	May 2010
Volume:	Vol.296
Number:	No.1
Number of Pages:	20
Page Range:	pp. 251-270
Identification Number:	10.1007/s00220-010-1015-x
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access
Funder:	Swedish Research Council, Japanese Ministry of Education, Culture, Sports, Science and Technology
Grant number:	18740086
URI:	http://wrap.warwick.ac.uk/id/eprint/6206

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