The $C^{1,\alpha }$ boundary Harnack principle in a slit domain and its application to the Signorini problem

We prove the $C^{2,\alpha }$ regularity of the free boundary in the Signorini problem with variable coefficients. We use a $C^{1,\alpha }$ boundary Harnack inequality in slit domain. The key method is to study a non-standard degenerate elliptic equation and obtain a $C^{1,\alpha }$ Schauder estimate.