The Scientific World Journal

Volume 2014 (2014), Article ID 875245, 6 pages

http://dx.doi.org/10.1155/2014/875245

Research Article

The Pólya-Szegö Principle and the Anisotropic Convex Lorentz-Sobolev Inequality

Shuai Liu and Binwu He

Department of Mathematics, Shanghai University, Shanghai 200444, China

Received 11 May 2014; Revised 24 June 2014; Accepted 24 June 2014; Published 15 July 2014

Academic Editor: Seyed S. Seyedalizadeh

Copyright © 2014 Shuai Liu and Binwu He. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

An anisotropic convex Lorentz-Sobolev inequality is established, which extends Ludwig, Xiao, and Zhang’s result to any norm from Euclidean norm, and the geometric analogue of this inequality is given. In addition, it implies that the (anisotropic) Pólya-Szegö principle is shown.

1. Introduction

The classical Pólya-Szegö principle (see, e.g., [1, 2]) states that for the inequality

holds for every , where denotes the set of functions on that are smooth and have compact support and is the standard Euclidean norm. Here denotes the Schwarz symmetrization of , that is, a function whose level sets have the same measure as the level sets of and are dilates of the Euclidean unit ball . It has important applications to a large class of variational problems in different areas, for example, isoperimetric inequalities, optimal forms of Sobolev inequalities, and sharp a priori estimates of solutions to second-order elliptic or parabolic boundary value problems.

An anisotropic version of the classical Pólya-Szegö principle has been proved in [3], where convex symmetrization of is involved, which states that if is an origin-symmetric compact convex set, then for the inequality

holds for every , where is the Minkowski functional of  the polar body of  . Here denotes the convex symmetrization of , that is, a function whose level sets have the same measure as the level sets of and are dilates of the set . Obviously, (2) reduces to (1) when (see Section 2 for unexplained notation and terminology).

A new approach to understanding Pólya-Szegö principle was proposed recently by Lutwak et al. [4] and Zhang [5]. Instead of using the classical technique on level sets , their approach is using the convexification of level sets . This technique plays a fundamental role in the newly emerged affine Pólya-Szegö principle (see, e.g., [4–9]). Despite this progress, the study of the Pólya-Szegö principle by using this technique is vacancy. This is the motivation of the present paper. More precisely, we show the Pólya-Szegö principle from the Brunn-Minkowski theory, different from the known proofs of the Pólya-Szegö principle based on the geometric measure theory (see, e.g., [1–3, 10–16]).

In [17], Ludwig et al. proved the following convex Lorentz-Sobolev inequality (see Theorem  2 in [17]): if and , then

where denotes the Lebesgue measure on with . This inequality has a geometric analogue, namely,  the following isoperimetric  inequality:  for ,

where is an origin-symmetric compact convex set in and is the surface area of .

In this paper we establish the following anisotropic convex Lorentz-Sobolev inequality.

Theorem 1. If , , and is an origin-symmetric convex body in , then

with equality if and only if is a dilate of for almost every .

Similarly, our inequality (5) has a geometric analogue, namely,  the following   Minkowski  inequality, for ,

where , are origin-symmetric compact convex sets in and is the mixed volume of , .

When , from , (5) and (6) reduce to (3) and (4), respectively.

It is shown that our inequality (5) implies the anisotropic Pólya-Szegö principle (2) for in Theorem 5. Hence it is also true in Euclidean case; that is, (3) implies (1) for . The arguments after Theorem 5 yield the fact that the anisotropic Pólya-Szegö principle (2) is still true for if we use the solution to the even normalized Minkowski problem.

2. Background Material

2.1. Elements of the Brunn-Minkowski Theory

For later reference, we quickly recall in this subsection some background material from the Brunn-Minkowski theory of convex bodies. This theory has its origin in the work of Firey from the 1960s and has expanded rapidly over the last couple of decades (see, e.g., [4, 8, 18–33]).

A convex body is a compact convex set in which is throughout assumed to contain the origin in its interior. We denote by the space of convex bodies equipped with the Hausdorff metric. Each convex body is uniquely determined by its support function defined by

Let denote the Minkowski functional of  ; that is, .

The polar set of is the convex body defined by

If , then it follows from the definitions of support functions and Minkowski functionals, as well as the definition of polar body, that

For ,  , the Minkowski combination is the convex body defined by

The mixed volume of   is defined in [25] by

In particular,

for every convex body .

It was shown in [25] that, for all convex bodies ,

where and the measure on is the classical surface area measure of . Recall that, for a Borel set , is the -dimensional Hausdorff measure of the set of all boundary points of for which there exists a normal vector of belonging to .

Note that

for all and convex bodies .

2.2. The Convex Symmetrization of Functions

Given any measurable function such that for every , its distribution function is defined by

The decreasing rearrangement of is defined by

The Schwarz symmetrization of is the function defined by

where is the standard Euclidean norm.

For an origin-symmetric convex body , the convex symmetrization of with respect to is defined as follows:

where is the Minkowski functional of , with being a dilate of so that . Note that ,  , and are equimeasurable; that is,

Therefore, we have

We will frequently apply Federer’s co-area formula (see, e.g., [34, page 258]). We state a version which is sufficient for our purposes: if is Lipschitz and is measurable, then, for any Borel set ,

where denotes -dimensional Hausdorff measure.

2.3. The Convexification of Level Sets

Suppose . For each real  , define the level set

By Sard’s theorem, for almost every , the boundary

of is a smooth -dimensional submanifold of with everywhere nonzero normal vector .

Now, we explain the technique called the convexification of  level sets (see [17] for more details). Let , where is open, be locally Lipschitz; let ; and suppose   for almost everywhere on  . For , define the convexification of the level set as the unique origin-symmetric convex body such that

for all even , where .

Thus, equality (24) holds for almost every if .

3. The Anisotropic Convex Lorentz-Sobolev Inequality

The following lemma can be proved in the spirit of [17, 31, 35](e.g., see Lemma  3 in [35]).

Lemma 2. If and are origin-symmetric convex bodies, then, for almost every and , is a dilate of and

Proof. Since is Lipschitz (and therefore differentiable almost everywhere) and on , then, for almost every ,

where is the outer unit normal vector of at the point . Note that , for almost every ; hence we have

Since is Lipschitz, then, for almost every , the set is the boundary of a dilate of with nonvanishing normal . It follows from Sard’s theorem that

Hence there exists a unique such that for almost every . Indeed, we have . Then by (24), (18), (9), and the fact that is homogeneous of degree and (27), we obtain that

for almost every and even . Thus, the uniqueness of the solution of the even Minkowski problem [25] and (14) implies that

where . Since for any and any , we have

for almost every .

Recall that the Minkowski inequality [25] states the following.

Theorem 3. If and , then

with equality if and only if  , are dilates when and if and only if , are homothetic when .

Now, we prove the anisotropic convex Lorentz-Sobolev inequality.

Proof of Theorem 1. Noting that , by the co-area formula (21), (24), (13), and (32), we have

where on for almost every and the second equality holds since is an origin-symmetric and the support function of is homogeneous of degree 1.

Equality (5) follows from equality (32) and the fact that is origin-symmetric.

It is shown above, Proof of Theorem 1, that the Minkowski inequality (32) implies inequality (5).

In what follows we will show that the Minkowski inequality (32) can be easily deduced from the anisotropic convex Lorentz-Sobolev inequality (5) for by taking

Indeed, as shown in [17, Lemma 8],

and with . Hence

where

4. The Pólya-Szegö Principle

The following theorem can be seen as a weak form of the Pólya-Szegö principle (2).

Theorem 4. If and , are origin-symmetric convex body such that is not a dilate of , then, for ,

Proof. Since is a dilate of for almost every by Lemma 2, then the Minkowski inequality (32) between and is strict for almost every . Combined with (25), it follows that

We are now in the position to prove the Pólya-Szegö principle (2).

Theorem 5. Suppose is an origin-symmetric convex bodies in . If , , then

Proof. It was shown in [4, (6.3)] that the following differential inequality holds:

Integrating both sides of the inequality gives

Noting that and Combined with (5), we obtain that

By the homogeneous of in (43) and (40), we only need to consider . So it is sufficient to prove that

The last equality is shown in [3]. Now, we prove this equality by using Lemma 2. Together with the co-area formula (21), the equality (24), the definition of in Lemma 2, (13), and , we obtain

where on for almost every . And the second equality holds since is origin-symmetric and the support function of is homogeneous of degree 1.

Moreover, Theorem 5 can be proved for by using the solution to the even normalized Minkowski problem as in [7, 9]. More precisely, suppose , for , and define the normalized convexification as the unique origin-symmetric convex body such that

for almost every . By taking slight modifications in the proof of Theorem 1, we obtain

Similar to the proof of Theorem 5, together with the observation in [7, (4.22)] that

we also get (43). So Theorem 5 remains true for .

Conflict of Interests

The authors declare that they have no conflict of interests.

Authors’ Contribution

All the authors contributed equally to the paper. All the authors read and approved the final paper.

Acknowledgments

This work is supported in part by the National Natural Science Foundation of China (Grant no. 11371239), Shanghai Leading Academic Discipline Project (Project no. J50101), and the Research Fund for the Doctoral Programs of Higher Education of China (20123108110001).

References

G. Pólya and G. Szegő, Isoperimetric Inequalities in Mathematical Physics, Princeton University Press, Princeton, NJ, USA, 1951.

G. Talenti, “Best constant in Sobolev inequality,” Annali di Matematica Pura ed Applicata. Serie Quarta, vol. 110, pp. 353–372, 1976.

A. Alvino, V. Ferone, and G. Trombetti, “Convex symmetrization and applications,” Annales de l'Institut Henri Poincare C: Non Linear Analysis, vol. 14, no. 2, pp. 275–293, 1997.

E. Lutwak, D. Yang, and G. Zhang, “Sharp affine Sobolev inequalities,” Journal of Differential Geometry, vol. 62, no. 1, pp. 17–38, 2002.

G. Zhang, “The affine Sobolev inequality,” Journal of Differential Geometry, vol. 53, no. 1, pp. 183–202, 1999.

D. Alonso-Gutiérrez, J. Bastero, and J. Bernués, “Factoring Sobolev inequalities through classes of functions,” Proceedings of the American Mathematical Society, vol. 140, no. 10, pp. 3557–3566, 2012.

A. Cianchi, E. Lutwak, D. Yang, and G. Zhang, “Affine Moser-Trudinger and MORrey-Sobolev inequalities,” Calculus of Variations and Partial Differential Equations, vol. 36, no. 3, pp. 419–436, 2009.

C. Haberl and F. E. Schuster, “Asymmetric affine Sobolev inequalities,” Journal of Functional Analysis, vol. 257, no. 3, pp. 641–658, 2009.

C. Haberl, F. Schuster, and J. Xiao, “An asymmetric affine Pólya-Szegö principle,” Mathematische Annalen, vol. 352, no. 3, pp. 517–542, 2012.

J. E. Brothers and W. P. Ziemer, “Minimal rearrangements of Sobolev functions,” Journal für die Reine und Angewandte Mathematik, vol. 384, pp. 153–179, 1988.

A. Cianchi, L. Esposito, N. Fusco, and T. Trombetti, “A quantitative Pólya-Szegö principle,” Journal für die Reine und Angewandte Mathematik, vol. 614, pp. 153–189, 2008.

A. Cianchi and N. Fusco, “Functions of bounded variation and rearrangements,” Archive for Rational Mechanics and Analysis, vol. 165, no. 1, pp. 1–40, 2002.

L. Esposito and P. Ronca, “Quantitative Pólya-Szegö principle for convex symmetrization,” Manuscripta Mathematica, vol. 130, no. 4, pp. 433–459, 2009.

L. Esposito and C. Trombetti, “Convex symmetrization and Pólya-Szegö inequality,” Nonlinear Analysis: Theory, Methods & Applications, vol. 56, no. 1, pp. 43–62, 2004.

A. Ferone and R. Volpicelli, “Minimal rearrangements of Sobolev functions: a new proof,” Annales de l'Institut Henri Poincare (C): Non Linear Analysis, vol. 20, no. 2, pp. 333–339, 2003.

A. Ferone and R. Volpicelli, “Convex rearrangement: equality cases in the Pólya-Szegö inequality,” Calculus of Variations and Partial Differential Equations, vol. 21, no. 3, pp. 259–272, 2004.

M. Ludwig, J. Xiao, and G. Zhang, “Sharp convex Lorentz-Sobolev inequalities,” Mathematische Annalen, vol. 350, no. 1, pp. 169–197, 2011.

S. Campi and P. Gronchi, “The -Busemann-Petty centroid inequality,” Advances in Mathematics, vol. 167, no. 1, pp. 128–141, 2002.

S. Campi and P. Gronchi, “On the reverse -Busemann-Petty centroid inequality,” Mathematika, vol. 49, no. 1-2, pp. 1–11 (2004), 2002.

C. Haberl and F. E. Schuster, “General Lp affine isoperimetric inequalities,” Journal of Differential Geometry, vol. 83, no. 1, pp. 1–26, 2009.

D. Hug, E. Lutwak, D. Yang, and G. Zhang, “On the Minkowski problem for polytopes,” Discrete & Computational Geometry, vol. 33, no. 4, pp. 699–715, 2005.

M. Ludwig, “Ellipsoids and matrix-valued valuations,” Duke Mathematical Journal, vol. 119, no. 1, pp. 159–188, 2003.

M. Ludwig, “Minkowski valuations,” Transactions of the American Mathematical Society, vol. 357, no. 10, pp. 4191–4213, 2005.

M. Ludwig and M. Reitzner, “A classification of invariant valuations,” Annals of Mathematics, vol. 172, no. 2, pp. 1219–1267, 2010.

E. Lutwak, “The Brunn-Minkowski-Firey theory. I. Mixed volumes and the Minkowski problem,” Journal of Differential Geometry, vol. 38, no. 1, pp. 131–150, 1993.

E. Lutwak, “The Brunn-Minkowski-Firey theory. {II}. Affine and geominimal surface areas,” Advances in Mathematics, vol. 118, no. 2, pp. 244–294, 1996.

E. Lutwak, D. Yang, and G. Zhang, “ affine isoperimetric inequalities,” Journal of Differential Geometry, vol. 56, no. 1, pp. 111–132, 2000.

E. Lutwak, D. Yang, and G. Zhang, “A new ellipsoid associated with convex bodies,” Duke Mathematical Journal, vol. 104, no. 3, pp. 375–390, 2000.

E. Lutwak, D. Yang, and G. Zhang, “On the Lp-Minkowski problem,” Transactions of the American Mathematical Society, vol. 356, no. 11, pp. 4359–4370, 2004.

E. Lutwak, D. Yang, and G. Zhang, “ John ellipsoids,” Proceedings of the London Mathematical Society, vol. 90, no. 2, pp. 497–520, 2005.

E. Lutwak, D. Yang, and G. Zhang, “Optimal Sobolev norms and the Minkowski problem,” International Mathematics Research Notices, vol. 2006, 21 pages, 2006.

E. Werner and D. Ye, “New affine isoperimetric inequalities,” Advances in Mathematics, vol. 218, no. 3, pp. 762–780, 2008.

E. Werner and D. Ye, “Inequalities for mixed -affine surface area,” Mathematische Annalen, vol. 347, no. 3, pp. 703–737, 2010.

H. Federer, Geometric Measure Theory, Springer, Berlin, Germany, 1969.

T. Wang, “The affine Pólya-Szegö principle: equality cases and stability,” Journal of Functional Analysis, vol. 265, no. 8, pp. 1728–1748, 2013.