# FRB

## Seminar

### Nondegeneracy in the Obstacle Problem with a Degenerate Force Term

Seminar Room 2, Newton Institute Gatehouse

#### Abstract

In this talk I present the proof of the optimal nondegeneracy of the solution $u$ of the obstacle problem $\triangle u=f\chi_{\{u>0\}}$ in a bounded domain $D\subset\mathbb{R}^{n}$, where we only require $f$ to have a nondegeneracy of the type $f(x)\geq\lambda\vert (x_1,\cdots,x_p)\vert^{\alpha}$ for some $\lambda>0$, $1\leq p\leq n$ (an integer) and $\alpha>0$. We prove optimal uniform $(2+\alpha)$-th order and nonuniform quadratic nondegeneracy, more precisely we prove that there exists $C>0$ (depending only on $n$, $p$ and $\alpha$) such that for $x$ a free boundary point and $r>0$ small enough we have $\sup_{\partial B_r(x)}u\geq C\lambda (r^{2+\alpha}+\vert(x_1,\cdots,x_p)\vert^{\alpha}r^{2})$. I also present the proof of the optimal growth with the assumption $\vert f(x)\vert\leq\Lambda\vert (x_1,\cdots,x_p)\vert^{\alpha}$ for some $\Lambda\geq 0$ and the porosity of the free boundary.Preprint: http://www.newton.ac.uk/preprints/NI14045.pdf