New monotonicity formulas and the optimal regularity in the Signorini problem with variable coefficients

We study the interior Signorini, or lower-dimensional obstacl problem for a uniformly elliptic divergence form operator L = div(A(x)del) with Lipschitz continuous coefficients. Our main result states that, similarly to what happens when L = Delta, the variational solution has the optimal interior regularity C-loc(1,1/2) (Omega(+/-) U M), when M is a codimension one fiat manifold which supports the obstacle. We achieve this by proving some new monotonicity formulas for an appropriate generalizatio of the celebrated Almgren's frequency functional. (C) 2014 Elsevier Inc. All rights reserved.