Electronic Journal of Differential Equations,
Vol. 2010(2010), No. 09, pp. 1-12.

Title: On the penalized obstacle problem in the unit half ball
       
Author: Erik Lindgren (CERMICS - ENPC, France)

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.
  
Submitted September 21, 2009. Published January 16, 2010.
Math Subject Classifications: 35J70, 35J60, 35J85
Key Words: Obstacle problem; elliptic equation; regularity; penalization.