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

In general, solutions $u$ to $$\mathrm{\Delta }u(\mathbf{x})=f(\mathbf{x}){\chi }_{\{u>\psi \}}$$ are not $C^{1,1}$, even for $f$ smooth and $\psi (\mathbf{x})\equiv 0$. 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 $\mathrm{\partial }\{u>\psi \}$, 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 $$\underset{j\to \mathrm{\infty }}{\lim }\frac{u(r_j\cdot +{\mathbf{x}}^0)}{\mathrm{\parallel }u{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(B_{r_j}({\mathbf{x}}^0))}},$$ $r_j\to 0^{+}$ are unique, the free boundary $\mathrm{\partial }\{u>\psi \}$ 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.