On the singularities of a free boundary through Fourier expansion

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.