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

Mixed norms, functional Inequalities, and Hamilton-Jacobi equations

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.
Citation: Antonio Avantaggiati, Paola Loreti, Cristina Pocci. Mixed norms, functional Inequalities, and Hamilton-Jacobi equations. Discrete and 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), 525-545. 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.

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), 525-545. 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.

[1]

Yasuhiro Fujita, Katsushi Ohmori. Inequalities and the Aubry-Mather theory of Hamilton-Jacobi equations. Communications on Pure and Applied Analysis, 2009, 8 (2) : 683-688. doi: 10.3934/cpaa.2009.8.683

[2]

Claudio Marchi. On the convergence of singular perturbations of Hamilton-Jacobi equations. Communications on Pure and Applied Analysis, 2010, 9 (5) : 1363-1377. doi: 10.3934/cpaa.2010.9.1363

[3]

Isabeau Birindelli, J. Wigniolle. Homogenization of Hamilton-Jacobi equations in the Heisenberg group. Communications on Pure and Applied Analysis, 2003, 2 (4) : 461-479. doi: 10.3934/cpaa.2003.2.461

[4]

Gonzalo Dávila. Comparison principles for nonlocal Hamilton-Jacobi equations. Discrete and Continuous Dynamical Systems, 2022  doi: 10.3934/dcds.2022061

[5]

Laura Caravenna, Annalisa Cesaroni, Hung Vinh Tran. Preface: Recent developments related to conservation laws and Hamilton-Jacobi equations. Discrete and Continuous Dynamical Systems - S, 2018, 11 (5) : i-iii. doi: 10.3934/dcdss.201805i

[6]

Fabio Camilli, Paola Loreti, Naoki Yamada. Systems of convex Hamilton-Jacobi equations with implicit obstacles and the obstacle problem. Communications on Pure and Applied Analysis, 2009, 8 (4) : 1291-1302. doi: 10.3934/cpaa.2009.8.1291

[7]

Olga Bernardi, Franco Cardin. On $C^0$-variational solutions for Hamilton-Jacobi equations. Discrete and Continuous Dynamical Systems, 2011, 31 (2) : 385-406. doi: 10.3934/dcds.2011.31.385

[8]

Emeric Bouin. A Hamilton-Jacobi approach for front propagation in kinetic equations. Kinetic and Related Models, 2015, 8 (2) : 255-280. doi: 10.3934/krm.2015.8.255

[9]

Gawtum Namah, Mohammed Sbihi. A notion of extremal solutions for time periodic Hamilton-Jacobi equations. Discrete and Continuous Dynamical Systems - B, 2010, 13 (3) : 647-664. doi: 10.3934/dcdsb.2010.13.647

[10]

Martino Bardi, Yoshikazu Giga. Right accessibility of semicontinuous initial data for Hamilton-Jacobi equations. Communications on Pure and Applied Analysis, 2003, 2 (4) : 447-459. doi: 10.3934/cpaa.2003.2.447

[11]

Xifeng Su, Lin Wang, Jun Yan. Weak KAM theory for HAMILTON-JACOBI equations depending on unknown functions. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6487-6522. doi: 10.3934/dcds.2016080

[12]

Gui-Qiang Chen, Bo Su. Discontinuous solutions for Hamilton-Jacobi equations: Uniqueness and regularity. Discrete and Continuous Dynamical Systems, 2003, 9 (1) : 167-192. doi: 10.3934/dcds.2003.9.167

[13]

David McCaffrey. A representational formula for variational solutions to Hamilton-Jacobi equations. Communications on Pure and Applied Analysis, 2012, 11 (3) : 1205-1215. doi: 10.3934/cpaa.2012.11.1205

[14]

Mihai Bostan, Gawtum Namah. Time periodic viscosity solutions of Hamilton-Jacobi equations. Communications on Pure and Applied Analysis, 2007, 6 (2) : 389-410. doi: 10.3934/cpaa.2007.6.389

[15]

Piermarco Cannarsa, Marco Mazzola, Carlo Sinestrari. Global propagation of singularities for time dependent Hamilton-Jacobi equations. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 4225-4239. doi: 10.3934/dcds.2015.35.4225

[16]

Qing Liu, Atsushi Nakayasu. Convexity preserving properties for Hamilton-Jacobi equations in geodesic spaces. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 157-183. doi: 10.3934/dcds.2019007

[17]

Olga Bernardi, Franco Cardin. Minimax and viscosity solutions of Hamilton-Jacobi equations in the convex case. Communications on Pure and Applied Analysis, 2006, 5 (4) : 793-812. doi: 10.3934/cpaa.2006.5.793

[18]

Kaizhi Wang, Jun Yan. Lipschitz dependence of viscosity solutions of Hamilton-Jacobi equations with respect to the parameter. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1649-1659. doi: 10.3934/dcds.2016.36.1649

[19]

Cui Chen, Jiahui Hong, Kai Zhao. Global propagation of singularities for discounted Hamilton-Jacobi equations. Discrete and Continuous Dynamical Systems, 2022, 42 (4) : 1949-1970. doi: 10.3934/dcds.2021179

[20]

Joan-Andreu Lázaro-Camí, Juan-Pablo Ortega. The stochastic Hamilton-Jacobi equation. Journal of Geometric Mechanics, 2009, 1 (3) : 295-315. doi: 10.3934/jgm.2009.1.295

2020 Impact Factor: 1.327

Metrics

  • PDF downloads (81)
  • HTML views (0)
  • Cited by (0)

[Back to Top]