The Pólya–Szegő inequality is a fundamental result in mathematical analysis concerning the behavior of Sobolev norms under symmetric decreasing rearrangement. It asserts that, for a suitable function, the Dirichlet integral (or more generally the $L^{p}$-norm of its gradient) does not increase when the function is replaced by its radially symmetric decreasing rearrangement.
Formal Statement
Let $ \Omega \subset \mathbb{R}^{n} $ be an open set (often taken as the whole space $ \mathbb{R}^{n} $ or a bounded domain) and let $ u \in W^{1,p}(\Omega) $ for $ 1 \le p < \infty $. Denote by $ u^{*} $ the symmetric decreasing (Schwarz) rearrangement of $ u $, i.e., the unique radially symmetric, non‑increasing function on a ball $ B \subset \mathbb{R}^{n} $ centered at the origin such that for every $ t > 0 $
$$ |{x \in \Omega : |u(x)| > t}| = |{x \in B : u^{*}(x) > t}|. $$
Then the Pólya–Szegő inequality states
$$ | abla u^{*}|{L^{p}(B)} \le | abla u|{L^{p}(\Omega)}. $$
In particular, for $ p=2 $,
$$ \int_{B} | abla u^{*}(x)|^{2},dx \le \int_{\Omega} | abla u(x)|^{2},dx . $$
The rearrangement also preserves the $L^{p}$-norm of the function itself:
$$ |u^{*}|{L^{p}(B)} = |u|{L^{p}(\Omega)} . $$
Historical Origin
The inequality is named after Gábor Pólya and Gábor Szegő, who presented it in their influential monograph Isoperimetric Inequalities in Mathematical Physics (1951). Their work built upon earlier concepts of symmetrization introduced by Schwarz and Steiner.
Equality Cases
Equality in the Pólya–Szegő inequality holds if and only if $ u $ is a translate of a radially symmetric decreasing function (modulo sets of measure zero) and, when the domain is a ball, when the level sets of $ u $ are concentric balls almost everywhere. Precise characterizations depend on the regularity of $ u $ and the exponent $ p $.
Extensions and Generalizations
- Weighted and anisotropic versions: Inequalities have been established for weighted Sobolev spaces and for rearrangements respecting anisotropic norms.
- Higher‑order Sobolev spaces: Analogous results exist for higher derivatives, often referred to as Pólya–Szegő–Talenti type inequalities.
- Manifolds and metric measure spaces: The principle extends, under suitable curvature or isoperimetric assumptions, to functions defined on Riemannian manifolds and more general metric spaces.
Applications
The Pólya–Szegő inequality is a key tool in several areas:
- Isoperimetric problems: It provides a method to compare energies of functions with their symmetrized counterparts, leading to proofs of optimal shapes (e.g., the Rayleigh–Faber–Krahn inequality for the first Dirichlet eigenvalue of the Laplacian).
- Sobolev embedding theorems: By reducing to radial functions, the inequality aids in establishing sharp constants in Sobolev and Gagliardo–Nirenberg inequalities.
- Partial differential equations: It is employed to obtain a priori estimates, symmetry results (via the method of moving planes), and to prove existence of ground states for nonlinear elliptic equations.
- Calculus of variations: In variational problems where the functional involves the Dirichlet integral, symmetrization arguments based on the Pólya–Szegő inequality often lead to minimizers that are radially symmetric.
References
- G. Pólya and G. Szegő, Isoperimetric Inequalities in Mathematical Physics, Princeton University Press, 1951.
- E. H. Lieb and M. Loss, Analysis, 2nd ed., American Mathematical Society, 2001 – Chapter on rearrangements.
- A. Baernstein, “Symmetrization in analysis”, Current Developments in Mathematics, 2000.
This entry reflects the established mathematical definition and significance of the Pólya–Szegő inequality as documented in standard analysis literature.