Hopf-Lax type formulas and hypercontractivity

Here we study u(x, t) := Qtu0(x)= y Rnu0(y)+α p 1-α pt→)q-1 H(y- e-α tx) where Han even, non negative, convex function, positively homogeneous of degree q as solution, in the viscosity sense, of an appropriate Hamilton-Jacobi equation. We show hypercontractivity and ultracontractivity inequalities, Logarithmic Sobolev Inequalities, Entropy-Energy inequality, and the optimality of the inequalities. © 2008 Università degli Studi di Napoli "Federico II".