On the singularities of a free boundary through Fourier expansion

Abstract: In this paper we are concerned with singular points of solutions to the {\it unstable} free boundary problem $$ \Delta u = - \chi_{{u>0}} \qquad \hbox{in} B_1. $$ 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 $\chi_{{u>0}} $. The method turns out to have enough momentum to accomplish a complete description of the structure of the singular set in ${\mathbb R}^3$. A surprising fact in ${\mathbb R}^3$ is that although $$\frac{u(r\x)}{\sup_{B_1}|u(r\x)|}$$ can converge at singularities to each of the harmonic polynomials $$ xy, {x^2+y^2\over 2}-z^2 \textrm{and} z^2-{x^2+y^2\over 2},$$ it may {\em not} converge to any of the non-axially-symmetric harmonic polynomials $\alpha((1+ \delta)x^2 +(1- \delta)y^2 - 2z^2)$ with $\delta\ne 1/2$. We also prove the existence of stable singularities in ${\mathbb R}^3$.