In mathematical analysis, the Pólya–Szegő inequality (or Szegő inequality) states that the Sobolev energy of a function in a Sobolev space does not increase under symmetric decreasing rearrangement. The inequality is named after the mathematicians George Pólya and Gábor Szegő.
Mathematical setting and statement Given a Lebesgue measurable function u : R n → R + , {\displaystyle u:\mathbb {R} ^{n}\to \mathbb {R} ^{+},} the symmetric decreasing rearrangement u ∗ : R n → R + , {\displaystyle u^{*}:\mathbb {R} ^{n}\to \mathbb {R} ^{+},} is the unique function such that for every t ∈ R , {\displaystyle t\in \mathbb {R} ,} the sublevel set u ∗
− 1 ( ( t , + ∞ ) ) {\displaystyle u^{*}{}^{-1}((t,+\infty ))} is an open ball centred at the origin 0 ∈ R n {\displaystyle 0\in \mathbb {R} ^{n}} that has the same Lebesgue measure as u − 1 ( ( t , + ∞ ) ) . {\displaystyle u^{-1}((t,+\infty )).} Equivalently, u ∗ {\displaystyle u^{*}} is the unique radial and radially nonincreasing function, whose strict sublevel sets are open and have the same measure as those of the function u {\displaystyle u} . The Pólya–Szegő inequality states that if moreover u ∈ W 1 , p ( R n ) , {\displaystyle u\in W^{1,p}(\mathbb {R} ^{n}),} then u ∗ ∈ W 1 , p ( R n ) {\displaystyle u^{*}\in W^{1,p}(\mathbb {R} ^{n})} and
∫ R n | ∇ u ∗ | p ≤ ∫ R n | ∇ u | p . {\displaystyle \int _{\mathbb {R} ^{n}}|\nabla u^{*}|^{p}\leq \int _{\mathbb {R} ^{n}}|\nabla u|^{p}.}
Applications of the inequality The Pólya–Szegő inequality is used to prove the Rayleigh–Faber–Krahn inequality, which states that among all the domains of a given fixed volume, the ball has the smallest first eigenvalue for the Laplacian with Dirichlet boundary conditions. The proof goes by restating the problem as a minimization of the Rayleigh quotient. The isoperimetric inequality can be deduced from the Pólya–Szegő inequality with p = 1 {\displaystyle p=1} . The optimal constant in the Sobolev inequality can be obtained by combining the Pólya–Szegő inequality with some integral inequalities.
Equality cases Since the Sobolev energy is invariant under translations, any translation of a radial function achieves equality in the Pólya–Szegő inequality. There are however other functions that can achieve equality, obtained for example by taking a radial nonincreasing function that achieves its maximum on a ball of positive radius and adding to this function another function which is radial with respect to a different point and whose support is contained in the maximum set of the first function. In order to avoid this obstruction, an additional condition is thus needed. It has been proved that if the function u {\displaystyle u} achieves equality in the Pólya–Szegő inequality and if the set { x ∈ R n : u ( x ) > 0 and ∇ u ( x ) = 0 } {\displaystyle \{x\in \mathbb {R} ^{n}:u(x)>0{\text{ and }}\nabla u(x)=0\}} is a null set for Lebesgue's measure, then the function u {\displaystyle u} is radial and radially nonincreasing with respect to some point a ∈ R n {\displaystyle a\in \mathbb {R} ^{n}} .
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