Analysis of the free boundary for the $p$-parabolic variational problem $(p\ge 2)$

Abstract

Variational inequalities (free boundaries), governed by the $p$-parabolic equation ($p\ge 2$), 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.