Ultracontractive bounds on Hamilton–Jacobi solutions

✍ Scribed by Ivan Gentil

• Publisher: Elsevier Science
• Year: 2002
• Tongue: French
• Weight: 151 KB
• Volume: 126
• Category: Article
• ISSN: 0007-4497
• DOI: 10.1016/s0007-4497(02)01128-4

✦ Synopsis

Following the equivalence between logarithmic Sobolev inequality, hypercontractivity of the heat semigroup showed by Gross and hypercontractivity of Hamilton-Jacobi equations, we prove, like the Varopoulos theorem, the equivalence between Euclidean-type Sobolev inequality and an ultracontractive control of the Hamilton-Jacobi equations. We obtain also ultracontractive estimations under general Sobolev inequality which imply in the particular case of a probability measure, transportation inequalities.