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

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.


Introduction
The classical Pólya-Szegö principle (see, e.g., [1,2]) states that for ≥ 1 the inequality holds for every ∈ ∞ 0 (R ), where ∞ 0 (R ) denotes the set of functions on R 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 ≥ 1 the inequality holds for every ∈ ∞ 0 (R ), 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).
In [17], Ludwig et al. proved the following convex Lorentz-Sobolev inequality (see Theorem 2 in [17]): if ∈ ∞ 0 (R ) and 1 ≤ < , then where denotes the Lebesgue measure on R with = ( ) = /2 /Γ(1 + /2). This inequality has a geometric analogue, namely, the following isoperimetric inequality: for 1 < < , 2 The Scientific World Journal where is an origin-symmetric compact convex set in R and ( ) is the surface area of . In this paper we establish the following anisotropic convex Lorentz-Sobolev inequality. Theorem 1. If ∈ ∞ 0 (R ), 1 ≤ ̸ = , and is an originsymmetric convex body in R , then with equality if and only if ⟨ ⟩ is a dilate of for almost every > 0.
It is shown that our inequality (5) implies the anisotropic Pólya-Szegö principle (2) for 1 ≤ ̸ = in Theorem 5. Hence it is also true in Euclidean case; that is, (3) implies (1) for 1 ≤ ̸ = . 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.
A convex body is a compact convex set in R which is throughout assumed to contain the origin in its interior. We denote by K the space of convex bodies equipped with the Hausdorff metric. Each convex body is uniquely determined by its support function ℎ = ℎ( , ⋅) : R → R defined by Let ‖ ⋅ ‖ : R → [0, ∞) denote the Minkowski functional of ∈ K ; that is, ‖ ‖ = min{ ≥ 0 : ∈ }. The polar set of ∈ K is the convex body defined by If ∈ K , then it follows from the definitions of support functions and Minkowski functionals, as well as the definition of polar body, that For ≥ 1, , ∈ K , the Minkowski combination + is the convex body defined by ℎ( + , ⋅) = ℎ( , ⋅) + ℎ( , ⋅) .
The mixed volume ( , ) of , ∈ K is defined in [25] by In particular, for every convex body . It was shown in [25] that, for all convex bodies , ∈ K , where ( , ) = ℎ ( ) 1− ( , ) and the measure ( , ⋅) on −1 is the classical surface area measure of . Recall that, for a Borel set Hausdorff measure of the set of all boundary points of for which there exists a normal vector of belonging to . Note that for all > 0 and convex bodies .

The Convex Symmetrization of Functions. Given any measurable function
The decreasing rearrangement * : The Schwarz symmetrization of is the function 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 of̃, with̃being a dilate of so that (̃) = . Note that , * , and are equimeasurable; that is, The Scientific World Journal 3 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 : R → R is Lipschitz and : R → [0, ∞) is measurable, then, for any Borel set ⊆ R, where H −1 denotes ( − 1)-dimensional Hausdorff measure.
By Sard's theorem, for almost every > 0, the boundary of [ ] is a smooth ( − 1)-dimensional submanifold of R with everywhere nonzero normal vector ∇ ( ). Now, we explain the technique called the convexification of level sets (see [17] for more details). Let :
Recall that the Minkowski inequality [25] states the following. Now, we prove the anisotropic convex Lorentz-Sobolev inequality.
It is shown above, Proof of Theorem 1, that the Minkowski inequality (32) implies inequality (5).

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

Theorem 5. Suppose is an origin-symmetric convex bodies in
The Scientific World Journal 5 Proof. It was shown in [4, (6.3)] that the following differential inequality holds: Integrating both sides of the inequality gives Noting that ℎ (⋅) = ‖ ⋅ ‖ 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) where ]( ) = −∇ ( )/|∇ ( )| on [ ] for almost every > 0. And the second equality holds since is originsymmetric and the support function of is homogeneous of degree 1.
Moreover, Theorem 5 can be proved for ≥ 1 by using the solution to the even normalized Minkowski problem as in [7,9]. More precisely, suppose ∈ ∞ 0 (R ), for ≥ 1, and define the normalized convexification⟨ ⟩ as the unique origin-symmetric convex body such that for almost every > 0. 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 = .