Abstract: This talk has two parts. In the first one, a brief description of local discontinuous Galerkin (LDG) method for Poisson problem will be given. Next, we propose and analyze the LDG method for an elliptic variational equality of the first kind that corresponds to a Poisson equation with Signorini type condition on part of the boundary. The method uses piecewise polynomials of degree one for the field variable and of degree zero or one for the approximation of its gradient. We show optimal convergence for the method and illustrate it with some numerical experiments.