TAILIEUCHUNG - Addition some inequality of volumes mixed in geometry

The issue norms for all volume preserving affine transformations for optimal Sobolev is always been interested in research. In this paper, we present some results of inequalities of volume mixed in geometry. | JOURNAL OF SCIENCE OF HNUE DOI Mathematical and Physical Sci. 2015 Vol. 60 No. 7 pp. 15-20 This paper is available online at http ADDITION SOME INEQUALITY OF VOLUMES MIXED IN GEOMETRY Lai Duc Nam Yen Bai Teacher s Training College Yen Bai Abstract. The issue norms for all volume preserving affine transformations for optimal Sobolev is always been interested in research. In this paper we present some results of inequalities of volume mixed in geometry. Keywords Lp Sobolev inequality geometry inequalities. 1. Introduction The classical sharp Lp Sobolev inequality states that if f W 1 p Rn with real p satisfying 1 p lt n then k f kp αn p kf k np n p where k kq denotes the usual Lq norm for functions on Rn . The optimal constants αn p in this inequality are due to Federer and Fleming 1 for p 1 and to Aubin 2 for 1 lt p lt n. For strengthened versions of see . 3 4 5 and the references therein. Recently Zhang 6 for p 1 and Lutwak Yang and Zhang 7 for 1 lt p lt n formulated and proved a sharp affine Lp Sobolev inequality. This remarkable inequality is invariant under all affine transformations of Rn while the classical Lp Sobolev inequality is invariant only under rigid motions. In the affine Lp Sobolev inequality the Lp norm of the Euclidean length of the gradient is replaced by an affine invariant of functions the Lp affine energy defined for f W 1 p Rn by Z 1 n Ep f cn p kDu f k n p du S n 1 nκ κ where cn p nκn 1 n 2κn p 2n p 1 1 p with κn π n 2 Γ 1 n2 and Du f is the directional derivative of f in the direction u. The sharp affine Lp Sobolev inequality of Zhang 6 and Lutwak Yang and Zhang 7 states that if f W 1 p Rn 1 p lt n then Ep f αn p kf k np n p Received October 10 2015. Accepted November 30 2015. Contact Lai Duc Nam e-mail address 15 Lai Duc Nam It is shown in 7 33 that k f kp Ep f . Hence the sharp affine Lp Sobolev inequality is stronger than the classical affine Lp

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