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Uniform Regularity Close to Cross Singularities in an Unstable Free Boundary Problem
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).ORCID iD: 0000-0002-1316-7913
2010 (English)In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 296, no 1, 251-270 p.Article in journal (Refereed) Published
Abstract [en]

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.

Place, publisher, year, edition, pages
2010. Vol. 296, no 1, 251-270 p.
Keyword [en]
composite membrane problem, asymptotics
National Category
Other Physics Topics
URN: urn:nbn:se:kth:diva-19325DOI: 10.1007/s00220-010-1015-xISI: 000275780200009ScopusID: 2-s2.0-77949655383OAI: diva2:337372
Swedish Research Council, 18740086
QC 20110114Available from: 2010-08-05 Created: 2010-08-05 Last updated: 2012-04-14Bibliographically approved

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Shahgholian, Henrik
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