We will describe the Signorini, or lower-dimensional obstacle problem, for a uniformly elliptic, divergence form operator L = div(A(x)\nabla) with Lipschitz continuous coefficients. We will give an overview of what is known about this problem, new developments and the role of a new monotonicity formula for an appropriate generalization of Almgren's frequency functional in the optimal regularity of the solution. Similarly to what happens in the Laplacian case, one of our main results states that the variational solution has the optimal interior regularity C^{1,1/2}.
This is joint work with Nicola Garofalo.