Nondegeneracy in the obstacle problem with a degenerate force term (Q500882)

In this interesting paper, the author proves optimal \((2+\alpha)\)-th order nondegeneracy of the solution \(u\) to the obstacle problem \(\Delta u=f\chi_{\{u>0\}}\), under the assumption that \(f\in L^{\infty}(D)\) and \(f(x)\geq \lambda |(x_1,\dots, x_p)|^{\alpha}\) for some \(\lambda >0\), \(1\leq p\leq n\) and \(\alpha>0\). More specifically, let \(D\subset \mathbb{R}^n\) be a bounded domain and consider the unique minimizer of the energy \[ \mathcal{E}(v):=\int_D \left(|\nabla v|^2+2fv\right)dx \] over the set \(\{v\in H^1(D) \;| \;v\geq 0 \text{ a.e. in } D \text{ and } v=g \text{ on } \partial D\}\), where \(g\in H^1(D)\). The main result of the paper states that there exists \(C=C(n,p,\alpha)>0\) such that if \(x^0\in\{x\in D \;| \;u(x)>0\}\) and \(B_r(x^0)\subset \subset D\), then \[ \sup_{\partial B_r(x^0)\cap \{x\in D \;| \;u(x)>0\}} u \geq u(x^0)+C\lambda r^2(r^{\alpha}+|(x_1^0,\dots,x_p^0)|^{\alpha}). \] Under the assumption that \(|f(x)|\leq \Lambda|(x_1,\dots,x_p)|^{\alpha}\), for some \(\Lambda\geq 0\), optimal growth is also obtained: if \(B_r(x^0)\subset D\), then \[ u(x)\leq C\left(u(x^0)+\Lambda r^2\left(r^{\alpha}+|(x_1^0,\dots,x_p^0)|^{\alpha}\right)\right). \] Finally, the free boundary \(\Gamma:=D\cap \partial \{x\in D \;| \;u(x)>0\}\) is proved to be locally porous in \(D\), under the assumption that \(\lambda|(x_1,\dots,x_p)|^{\alpha}\leq f(x)\leq \Lambda|(x_1,\dots, x_p)|^{\alpha}\) for every \(x\in D\).