Obstacle problem with a degenerate force term

Abstract

We study the regularity of the free boundary at its intersection with the line $\left\{x_1=0\right\}$ in the obstacle problem

$$△u=\left|x_1\right|{\chi }_{\left\{u>0\right\}}\text{ in }D,$$

where $D\subset {ℝ}^2$ is a bounded domain such that $D\cap \left\{x_1=0\right\}\ne \varnothing$.

We obtain a uniform $C^{1,1}$ bound on cubic blowups; we find all homogeneous global solutions; we prove the uniqueness of the blowup limit; we prove the convergence of the free boundary to the free boundary of the blowup limit; at the points with lowest Weiss balanced energy, we prove the convergence of the normal of the free boundary to the normal of the free boundary of the blowup limit and that locally the free boundary is a graph; and, finally, for a particular case we prove that the free boundary is not $C^{1,\alpha }$ regular near to a degenerate point for any $0<\alpha <1$.