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

Abstract

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