Abstract

In this talk I present the proof of the optimal nondegeneracy of the solution $u$ of the obstacle problem $ riangle u=fi_{{u>0}}$ in a bounded domain $D ubsetmathbb{R}^{$, where we only require $f$ to have a nondegeneracy of the type $f(x)geqlambdaert (x_1,ots,x_p)ert}{lpha}$ for some $lambda>0$, $1leq pleq n$ (an integer) and $lpha>0$. We prove optimal uniform $(2+lpha)$-th order and nonuniform quadratic nondegeneracy, more precisely we prove that there exists $C>0$ (depending only on $n$, $p$ and $lpha$) such that for $x$ a free boundary point and $r>0$ small enough we have $ up_{partial B_r(x)}ugeq Clambda (r^{+ert(x_1,ots,x_p)ert}{lpha}r^{)$. I also present the proof of the optimal growth with the assumption $ert f(x)ertleqLambdaert (x_1,ots,x_p)ert}{lpha}$ for some $Lambdageq 0$ and the porosity of the free boundary.

Preprint: http://www.newton.ac.uk/preprints/NI14045.pdf