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On the singularities of a free boundary through Fourier expansion
Mathematics Institute, University of Warwick.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).ORCID iD: 0000-0002-1316-7913
Mathematical Institute of the Heinrich Heine University.
2012 (English)In: Inventiones Mathematicae, ISSN 0020-9910, E-ISSN 1432-1297, Vol. 187, no 3, 535-587 p.Article in journal (Refereed) Published
Abstract [en]

In this paper we are concerned with singular points of solutions to the unstable free boundary problem Delta u = -chi({u>0}) in B-1. The problem arises in applications such as solid combustion, composite membranes, climatology and fluid dynamics. It is known that solutions to the above problem may exhibit singularities-that is points at which the second derivatives of the solution are unbounded-as well as degenerate points. This causes breakdown of by-now classical techniques. Here we introduce new ideas based on Fourier expansion of the non-linearity chi({u>0}). The method turns out to have enough momentum to accomplish a complete description of the structure of the singular set in R-3. A surprising fact in R-3 is that although u(rx)/sup(B1) vertical bar u(rx)vertical bar can converge at singularities to each of the harmonic polynomials xy, x(2) + y(2)/2 - z(2) and z(2) - x(2) + y(2)/2, it may not converge to any of the non-axially-symmetric harmonic polynomials alpha((1 + delta)x(2) + (1 - delta)y(2) - 2z(2)) with delta not equal 1/2. We also prove the existence of stable singularities in R-3.

Place, publisher, year, edition, pages
2012. Vol. 187, no 3, 535-587 p.
National Category
URN: urn:nbn:se:kth:diva-82114DOI: 10.1007/s00222-011-0336-5ISI: 000302695800002ScopusID: 2-s2.0-84857796756OAI: diva2:497943
Swedish Research Council
QC 20120509Available from: 2012-02-11 Created: 2012-02-11 Last updated: 2012-05-09Bibliographically approved

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Shahgholian, Henrik
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