Sharp Hardy inequalities in the half space with trace remainder term

We deal with a class of inequalities which interpolate the Kato's inequality and the Hardy's inequality in the half space. Starting from the classical Hardy's inequality in the half space R^n_+ =R^{n-1}x(0,infty ), we show that, if we replace the optimal constant (n-2)^2/4 with a smaller one (c-2)^2/4 , 2\le c< n, then we can add an extra trace-term equals to that one that appears in the Kato's inequality. The constant in the trace remainder term is optimal and it tends to zero when c goes to n, while it is equal to the optimal constant in the Kato's inequality when c=2. The approach is based on a very classical method of Calculus of Variation due to Weirstrass (and developed by Hilbert) that usually is considered to prove that the solutions of the Euler Lagrange equation associated to a functional are, in fact, extremals. In this paper we will show how this method is well suited also to functionals that have no extremals.