On the penalized obstacle problem in the unit half ball

Abstract

We study the penalized obstacle problem in the unit half ball, i.e. an approximation of the obstacle problem in the unit half ball. The main result states that when the approximation parameter is small enough and when certain level sets are sufficiently close to the hyperplane ${x_1=0}$, then these level sets are uniformly $C^1$ regular graphs. As a by-product, we also recover some regularity of the free boundary for the limiting problem, i.e., for the obstacle problem.

Keywords

• Obstacle problem
• elliptic equation
• regularity
• penalization