Hopf-Lax type formulas and hypercontractivity

Abstract

Here we study $$u(x, t) := Q_{t}u_0(x)=\min_{y \in R^n}\left\{u_{0}(y)+\left({{\alpha p}\over{ 1-e^{-\alpha pt}}}\right)^{q-1} H^\star(y- e^{-\alpha t}x)\right\}$$ where H* an 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.