Gagliardo-Nirenberg inequalities involving the gradient L2-norm

We present a method giving the sharp constants and optimal functions of all the Gagliardo-Nirenberg inequalities involving the L-2-norm of the gradient. We show that the optimal functions can be explicitly derived from a specific non-linear ordinary differential equation which appears to be linear for a subclass of the Gagliardo-Nirenberg inequalities or when the space dimension reduces to 1. In these cases, we give the explicit expressions of the optimal functions, along with the sharo constants of the corresponding Gagliardo-Nirenberg inequalities. Our method extend to the L-p-Gagliardo-Nirenberg and L-p-Nash's inequalities for all p > 1.