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Abstract Variational inequalities (free boundaries), governed by the p-parabolic equation (p > 2), are the objects of investigation in this paper. Using intrinsic scaling we establish the behavior of solutions near the free boundary. A consequence of this is that the time levels of the free boundary are porous (in N-dimension) and therefore its Hausdorff dimension is less than N. In particular the N-Lebesgue measure of the free boundary is zero for each t-level.

Let $D$ be either the unit ball $B_1\left(0\right)$ or the half ball $B_1^{+}\left(0\right),$ let $f$ be a strictly positive and continuous function, and let $u$ and $\Omega \subset D$ solve the following overdetermined problem: $$\Delta u\left(x\right)={\chi }_{{}_{\Omega }}\left(x\right)f\left(x\right)\text{in}D,0\in \partial \Omega ,u=\left|\nabla u\right|=0\text{in}{\Omega }^c,$$ where ${\chi }_{{}_{\Omega }}$ denotes the characteristic function of $\Omega ,$ ${\Omega }^c$ denotes the set $D\setminus \Omega ,$ and the equation is satisfied in the sense of distributions. When $D=B_1^{+}\left(0\right),$ then we impose in addition that $$u\left(x\right)\equiv 0\text{on}\left\{(x^{\text{'}},x_n)\right|x_n=0\}.$$ We show that a fairly mild thickness assumption on ${\Omega }^c$ will ensure enough compactness on $u$ to give us...