Classification of Blow-ups and Free Boundaries of Solutions to Unstable Free Boundary Problems

In general, solutions $u$ to \[ \Delta u(\mathbf{x})=f(\mathbf{x})\chi_{\{u>\psi\}} \] are not $C^{1,1}$, even for $f$ smooth and $\psi(\mathbf{x})\equiv0$. Points around which $u$ is not $C^{1,1}$ are called singular points, and the set of all such points, the singular set. In this article we analyze blow-ups, the free boundary $\partial\{u>\psi\}$, and the singular set close to singular points $\mathbf{x}^{0}=(x^{0},y^{0},z^{0})$ in $\mathbb{R}^{3}$. We show that blow-ups of the form \[ \lim_{j\to\infty}\frac{u(r_{j}\cdot+\mathbf{x}^{0})}{\|u\|_{L^{\infty}(B_{r_{j}}(\mathbf{x}^{0}))}}, \] $r_{j}\to0^{+}$ are unique, the free boundary $\partial\{u>\psi\}$ is up to rotations close to the surfaces $(x-x^{0})^{2}+(y-y^{0})^{2}=2(z-z^{0})^{2}$ or $(x-x^{0})^{2}=(z-z^{0})^{2}$, and that singular points are either isolated or contained in a $C^{1}$ curve. The methods of the proofs are based on projecting the solutions $u$ on the space of harmonic two-homogeneous polynomials.