Optimal regularity for the Signorini problem and its free boundary

We will show optimal regularity for minimizers of the Signorini problem for the Lame system. In particular if u = (u(1), u(2), ..., u(n)) is an element of W-1,W-2(B-1(+) : R-n) minimizes J (u) = integral(B1+) vertical bar del u+del(perpendicular to)vertical bar(2) + lambda div(u)(2) in the convex set K = {u = (u(1), u(2), ... , u(n)) is an element of W-1,W-2 (B-1(+) : R-n); u(n) = 0 on Pi, u = f is an element of C-infinity(partial derivative B-1) on (partial derivative B-1)(+)}, where lambda >= 0 say. Then u is an element of C(1,1/)2(B-1/2(+)). Moreover the free boundary, given by Gamma(u) = partial derivative{x; u(n) (x) = 0, x(n) = 0} boolean AND B-1, will be a C-1,C-alpha graph close to points where u is not degenerate. Similar results have been know before for scalar partial differential equations (see for instance [5,6]). The novelty of this approach is that it does not rely on maximum principle methods and is therefore applicable to systems of equations.