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 & 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

[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

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

[1]

Yasuhiro Fujita, Katsushi Ohmori. Inequalities and the Aubry-Mather theory of Hamilton-Jacobi equations. Communications on Pure & 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 & 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 & Applied Analysis, 2003, 2 (4) : 461-479. doi: 10.3934/cpaa.2003.2.461

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

Martino Bardi, Yoshikazu Giga. Right accessibility of semicontinuous initial data for Hamilton-Jacobi equations. Communications on Pure & 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 & Continuous Dynamical Systems - A, 2016, 36 (11) : 6487-6522. doi: 10.3934/dcds.2016080

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

Jochen Merker. Generalizations of logarithmic Sobolev inequalities. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 329-338. doi: 10.3934/dcdss.2008.1.329

[20]

Tomoki Ohsawa, Anthony M. Bloch. Nonholonomic Hamilton-Jacobi equation and integrability. Journal of Geometric Mechanics, 2009, 1 (4) : 461-481. doi: 10.3934/jgm.2009.1.461

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (0)

[Back to Top]