Mixed norms, functional Inequalities, and Hamilton-Jacobi equations
Antonio Avantaggiati, Paola Loreti, Cristina Pocci · Discrete and Continuous Dynamical Systems - B · 2014
In this paper we generalize the notion of hypercontractivity for nonlinear semigroups allowing the functions to belong to mixed spaces. As an application of this notion, we consider a class of Hamilton-Jacobi equations and we establish functional inequalities. More precisely, we get hypercontractivity for viscosity solutions given in terms of Hopf-Lax type formulas. In this framework, we consider different measures associated with the variables; consequently, using mixed norms, we find new inequalities.The novelty of this approach is the study of functional inequalities with mixed norms for semigroups.