Article Contents
Article Contents

# Mixed norms, functional Inequalities, and Hamilton-Jacobi equations

• 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.
Mathematics Subject Classification: Primary: 47A30, 35F20; Secondary: 26D15.

 Citation:

•  [1] A. Avantaggiati and P. Loreti, Hypercontractivity, Hopf-Lax type formulas, Ornstein-Uhlenbeck operators. II, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 525-545.doi: 10.3934/dcdss.2009.2.525. [2] A. Avantaggiati, P. Loreti and C. Pocci, On a class of Hamilton-Jacobi equations with related logarithmic Sobolev inequality, and optimality, Commun. Appl. Ind. Math., 2 (2011), 1-16.doi: 10.1685/journal.caim.389. [3] A. Benedek and R. Panzone, The space $L^p$, with mixed norm, Duke Math. J., 28 (1961), 301-324.doi: 10.1215/S0012-7094-61-02828-9. [4] S. G. Bobkov, I. Gentil and M. Ledoux, Hypercontractivity of Hamilton-Jacobi equations, J. Math. Pures Appl. (9), 80 (2001), 669-696.doi: 10.1016/S0021-7824(01)01208-9. [5] S. G. Bobkov and F. Götze, Exponential integrability and transportation cost related to logarithmic Sobolev inequalities, J. Funct. Anal., 163 (1999), 1-28.doi: 10.1006/jfan.1998.3326. [6] I. Gentil, Ultracontractive bounds on Hamilton-Jacobi solutions, Bull. Sci. Math., 126 (2002), 507-524.doi: 10.1016/S0007-4497(02)01128-4. [7] L. Gross, Logarithmic Sobolev inequalities, Amer. J. Math., 97 (1975), 1061-1083.doi: 10.2307/2373688. [8] H.-J. Schmeisser and H. Triebel, Topics in Fourier Analysis and Function Spaces, A Wiley-Interscience Publication, John Wiley & Sons Ltd., Chichester, 1987.