Andersson, John Erik, Shahgholian, Henrik and Weiss, Georg S.. (2012) On the singularities of a free boundary through Fourier expansion. Inventiones Mathematicae, Vol.187 (No.3). pp. 535-587. ISSN 0020-9910

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Abstract

In this paper we are concerned with singular points of solutions to the unstable free boundary problem u=−u0in B1 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 nonlinearity χ {u>0}. The method turns out to have enough momentum to accomplish a complete description of the structure of the singular set in ℝ3. A surprising fact in ℝ3 is that although u(rx)supB1u(rx) can converge at singularities to each of the harmonic polynomials xy2x2+y2−z2andz2−2x2+y2 it may not converge to any of the non-axially-symmetric harmonic polynomials α((1+δ)x 2+(1−δ)y 2−2z 2) with δ≠1/2. We also prove the existence of stable singularities in ℝ3.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Mathematics
Library of Congress Subject Headings (LCSH):	Fourier series, Boundary value problems
Journal or Publication Title:	Inventiones Mathematicae
Publisher:	Springer
ISSN:	0020-9910
Date:	March 2012
Volume:	Vol.187
Number:	No.3
Page Range:	pp. 535-587
Identification Number:	10.1007/s00222-011-0336-5
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access
URI:	http://wrap.warwick.ac.uk/id/eprint/39060

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