Nondegeneracy in the obstacle problem with a degenerate force term

Abstract

In this paper we prove the optimal non degeneracy of the solution $u$ of the obstacle problem $△u=f{\chi }_{\{u>0\}}$ in a bounded domain $D\subset {\mathbb{R}}^n$, where we only require $f$ to have a nondegeneracy of the type $f(x)\ge \lambda |(x_1,\cdots ,x_p){|}^{\alpha }$ for some $\lambda >0$, $1\le p\le n$ (an integer) and $\alpha >0$. We prove optimal $(2+\alpha )$-th order non degeneracy. We also prove the optimal growth with the assumption $|f(x)|\le \Lambda |(x_1,\cdots ,x_p){|}^{\alpha }$ for some $\Lambda \ge 0$ and the porosity of the free boundary.