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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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[6]

I. Gentil, Ultracontractive bounds on Hamilton-Jacobi solutions,, Bull. Sci. Math., 126 (2002), 507.  doi: 10.1016/S0007-4497(02)01128-4.  Google Scholar

[7]

L. Gross, Logarithmic Sobolev inequalities,, Amer. J. Math., 97 (1975), 1061.  doi: 10.2307/2373688.  Google Scholar

[8]

H.-J. Schmeisser and H. Triebel, Topics in Fourier Analysis and Function Spaces,, A Wiley-Interscience Publication, (1987).   Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

I. Gentil, Ultracontractive bounds on Hamilton-Jacobi solutions,, Bull. Sci. Math., 126 (2002), 507.  doi: 10.1016/S0007-4497(02)01128-4.  Google Scholar

[7]

L. Gross, Logarithmic Sobolev inequalities,, Amer. J. Math., 97 (1975), 1061.  doi: 10.2307/2373688.  Google Scholar

[8]

H.-J. Schmeisser and H. Triebel, Topics in Fourier Analysis and Function Spaces,, A Wiley-Interscience Publication, (1987).   Google Scholar

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