University of Cambridge > Talks.cam > Isaac Newton Institute Seminar Series > Nondegeneracy in the Obstacle Problem with a Degenerate Force Term

Nondegeneracy in the Obstacle Problem with a Degenerate Force Term

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Free Boundary Problems and Related Topics

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 $f(x)geqlambda ert (x_1,ots,x_p) ert{lpha}$ 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) ertleqLambda ert (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

This talk is part of the Isaac Newton Institute Seminar Series series.

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