Optimal regularity for the Signorini problem

We prove under general assumptions that solutions of the thin obstacle or Signorini problem in any space dimension achieve the optimal regularity C (1,1/2). This improves the known optimal regularity results by allowing the thin obstacle to be defined in an arbitrary C (1,beta) hypersurface, beta > 1/2, additionally, our proof covers any linear elliptic operator in divergence form with smooth coefficients. The main ingredients of the proof are a version of Almgren's monotonicity formula and the optimal regularity of global solutions.