Lagrange multipliers and characteristic functions

We consider a stationary variational inequality with gradient constraint and obstacle. We prove that this problem can be described by an equation using a Lagrange multiplier and a characteristic function. The Lagrange multiplier contains information about the contact set of the modulus of the gradient of the solution with the gradient constraint, and the characteristic function is defined in the contact set of the solution with the obstacle. Moreover, given a convergent sequence of data, we prove the stability of the corresponding solutions.