Properties of the free boundaries for the obstacle problem of the porous medium equations

In this paper, we study the existence and interior W-2,W-p-regularity of the solution, and the regularity of the free boundary partial derivative{u > phi} to the obstacle problem of the porous medium equation, u(t) = Delta u(m) (m > 1) with the obstacle function phi. The penalization method is applied to have the existence and interior regularity. To deal with the interaction between two free boundaries partial derivative{u > phi} and partial derivative{u > 0}, we consider two cases on the initial data which make the free boundary partial derivative{u > phi} separate from the free boundary partial derivative{u > 0}. Then the problem is converted into the obstacle problem for a fully nonlinear operator. Hence, the C-1-regularity of the free boundary partial derivative(u > phi} is obtained by the regularity theory of a class of obstacle problems for the general fully nonlinear operator.