Global solutions of the obstacle problem in half-spaces, and their impact on local stability

We show that there are an abundance of non-homogeneous global solutions to the obstacle problem, in the half-space, with a (fixed) homogeneous boundary condition As a consequence we obtain local instability of the free boundary under C^1,1 perturbation, of the Dirichlet data. Received: 15 December 2003, Accepted: 7 July 2004, Published online: 8 February 2005 Mathematics Subject Classification (2000): 35R35, 34D10 H. Shahgholian was supported in part by Swedish Research Council.