From the Uniform Approximation of a Solution of the PDE to the L ² -Approximation of the Gradient of the Solution

We establish a new energy inequality for the difference of two semiconvex functions in a bounded open convex set D of

. This inequality is applied to the

L^2 L 2

-approximation problem of the gradient of the unknown solution of the nonlinear elliptic partial differential equation provided that the latter solution is a semiconvex function in D.