, Volume 187, Issue 3, pp 535-587
Date: 09 Jun 2011

On the singularities of a free boundary through Fourier expansion

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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.

H. Shahgholian has been supported in part by the Swedish Research Council. G.S. Weiss has been partially supported by the Grant-in-Aid 21540211 of the Japanese Ministry of Education, Culture, Sports, Science and Technology. He also thanks the Knut och Alice Wallenberg foundation for a visiting appointment to KTH. Both J. Andersson and G.S. Weiss thank the Göran Gustafsson Foundation for visiting appointments to KTH.