The Library
On the singularities of a free boundary through Fourier expansion
Tools
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. doi:10.1007/s00222-011-0336-5 ISSN 0020-9910.
Research output not available from this repository.
Request-a-Copy directly from author or use local Library Get it For Me service.
Official URL: http://dx.doi.org/10.1007/s00222-011-0336-5
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, Engineering and Medicine > Science > Mathematics | ||||
Library of Congress Subject Headings (LCSH): | Fourier series, Boundary value problems | ||||
Journal or Publication Title: | Inventiones Mathematicae | ||||
Publisher: | Springer | ||||
ISSN: | 0020-9910 | ||||
Official Date: | March 2012 | ||||
Dates: |
|
||||
Volume: | Vol.187 | ||||
Number: | No.3 | ||||
Page Range: | pp. 535-587 | ||||
DOI: | 10.1007/s00222-011-0336-5 | ||||
Status: | Peer Reviewed | ||||
Publication Status: | Published | ||||
Access rights to Published version: | Restricted or Subscription Access |
Request changes or add full text files to a record
Repository staff actions (login required)
View Item |