Mixed norms, functional Inequalities, and Hamilton-Jacobi equations

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September 2014, 19(7): 1855-1867. doi: 10.3934/dcdsb.2014.19.1855

Mixed norms, functional Inequalities, and Hamilton-Jacobi equations

Antonio Avantaggiati ^1, , Paola Loreti ^1, and Cristina Pocci ^2,

1.	Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Sezione di Matematica, Sapienza Università di Roma, Via A. Scarpa 16, 00161 Roma, Italy, Italy
2.	Dipartimento di Scienze di Base e Applicate, per l'Ingegneria-Sezione di Matematica, Sapienza Università di Roma, Via A. Scarpa 16, 00161 Roma

Received April 2013 Revised February 2014 Published August 2014

Abstract
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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.

Keywords: logarithmic Sobolev inequalities., hypercontractivity, Mixed norms, Hamilton-Jacobi equations.

Mathematics Subject Classification: Primary: 47A30, 35F20; Secondary: 26D1.

Citation: Antonio Avantaggiati, Paola Loreti, Cristina Pocci. Mixed norms, functional Inequalities, and Hamilton-Jacobi equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 1855-1867. doi: 10.3934/dcdsb.2014.19.1855

References:

[1]	A. Avantaggiati and P. Loreti, Hypercontractivity, Hopf-Lax type formulas, Ornstein-Uhlenbeck operators. II,, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 525. 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. 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. 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. 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. doi: 10.1006/jfan.1998.3326.
[6]	I. Gentil, Ultracontractive bounds on Hamilton-Jacobi solutions,, Bull. Sci. Math., 126 (2002), 507. doi: 10.1016/S0007-4497(02)01128-4.
[7]	L. Gross, Logarithmic Sobolev inequalities,, Amer. J. Math., 97 (1975), 1061. doi: 10.2307/2373688.
[8]	H.-J. Schmeisser and H. Triebel, Topics in Fourier Analysis and Function Spaces,, A Wiley-Interscience Publication, (1987).

show all references

References:

[1]	A. Avantaggiati and P. Loreti, Hypercontractivity, Hopf-Lax type formulas, Ornstein-Uhlenbeck operators. II,, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 525. 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. 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. 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. 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. doi: 10.1006/jfan.1998.3326.
[6]	I. Gentil, Ultracontractive bounds on Hamilton-Jacobi solutions,, Bull. Sci. Math., 126 (2002), 507. doi: 10.1016/S0007-4497(02)01128-4.
[7]	L. Gross, Logarithmic Sobolev inequalities,, Amer. J. Math., 97 (1975), 1061. doi: 10.2307/2373688.
[8]	H.-J. Schmeisser and H. Triebel, Topics in Fourier Analysis and Function Spaces,, A Wiley-Interscience Publication, (1987).

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