Article

Inventiones mathematicae

, Volume 187, Issue 3, pp 535-587

First online:

On the singularities of a free boundary through Fourier expansion

  • John AnderssonAffiliated withMathematics Institute, University of Warwick
  • , Henrik ShahgholianAffiliated withDepartment of Mathematics, Royal Institute of Technology Email author 
  • , Georg S. WeissAffiliated withMathematical Institute of the Heinrich Heine University

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Abstract

In this paper we are concerned with singular points of solutions to the unstable free boundary problem
$$\Delta u = - \chi_{\{u>0\}} \quad\hbox{in } B_1.$$
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 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
$$\frac{u(r\mathbf{x})}{\sup_{B_1}|u(r\mathbf{x})|}$$
can converge at singularities to each of the harmonic polynomials
$$xy,\qquad {x^2+y^2\over2}-z^2\quad \textrm{and}\quad z^2-{x^2+y^2\over2},$$
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.

Mathematics Subject Classification (2000)

35R35 35B40 35J60