Mixed norms, functional Inequalities, and Hamilton-Jacobi equations

Antonio Avantaggiati; Paola Loreti; Cristina Pocci

doi:10.3934/dcdsb.2014.19.1855

Article Contents

2014, Volume 19, Issue 7: 1855-1867. Doi: 10.3934/dcdsb.2014.19.1855

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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
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: March 31, 2013

Revised: January 31, 2014

Published: August 2014

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Abstract

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.

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

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References

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