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 $$\Delta u=-{\chi }_{\{u>0\}}in{B}_1.$$ 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 $$xy,\frac{x^2+y^2}{2}-z^2\text{and}{z}^2-\frac{x^2+y^2}{2},$$ 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 ${\mathbb{R}}^3$.

39 pages, 5 figures

References in corpus (1)

• Cross-shaped and Degenerate Singularities in an Unstable Elliptic Free Boundary Problem