Boundary estimates for solutions of the obstacle problem with two phases

Abstract

For weak solutions of the two-phase obstacle problem \Delta u=\lambda^{+}\chi_{\{u>0\}}-\lambda^{-}\chi_{\{u<0\}} in B_{1}^{+}, \lambda^{\pm}\geq0, \lambda^{+}+\lambda^{-}>0, satisfying the non-zero Dirichlet condition on the flat part of \partial B_{1}^{+}, we obtain the optimal regularity, i.e., we show that u is a W_{\infty}^{2}-function.