Reflected BSDE's , PDE's and Variational Inequalities

We discuss a class of semilinear PDE's with obstacle, of the form (_t+L)u+f(t,- x,u,^*u)+=0,uh,u_T=g where h is the obstacle. The solution of such an equation (in variational sense) is a couple (u,) where uL^2([0,T];H^1) and is a positive Radon measure concentrated on u=h. We prove that this equation has a unique solution and u is the maximal solution of the correspond- ing variational inequality. The probabilistic interpretation (Feynman-Kac formula) is given by means of Reflected Backward Stochastic Differential Equations. We give a new construction of solutions of such equations using a maximum principle. This perimts to consider obstacles with jumps