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

Abstract

We study the interior Signorini, or lower-dimensional obstacle problem for a uniformly elliptic divergence form operator L=div(A(x)∇) with Lipschitz continuous coefficients. Our main result states that, similarly to what happens when L=Δ, the variational solution has the optimal interior regularity Cloc1,12(Ω±∪M), when M is a codimension one flat manifold which supports the obstacle. We achieve this by proving some new monotonicity formulas for an appropriate generalization of the celebrated Almgren's frequency functional.