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.
[1] The inequality is named after the mathematicians George Pólya and Gábor Szegő.
Given a Lebesgue measurable function
the symmetric decreasing rearrangement
is an open ball centred at the origin
is the unique radial and radially nonincreasing function, whose strict sublevel sets are open and have the same measure as those of the function
and 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.
[2][3] 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.
achieves equality in the Pólya–Szegő inequality and if the set
is a null set for Lebesgue's measure, then the function
[4] The Pólya–Szegő inequality is still valid for symmetrizations on the sphere or the hyperbolic space.
[5] The inequality also holds for partial symmetrizations defined by foliating the space into planes (Steiner symmetrization)[6][7] and into spheres (cap symmetrization).
[8][9] There are also Pólya−Szegő inequalities for rearrangements with respect to non-Euclidean norms and using the dual norm of the gradient.
[10][11][12] The original proof by Pólya and Szegő for
was based on an isoperimetric inequality comparing sets with cylinders and an asymptotics expansion of the area of the area of the graph of a function.
[1] The inequality is proved for a smooth function
that vanishes outside a compact subset of the Euclidean space
They use then the geometrical fact that since the horizontal slices of both sets have the same measure and those of the second are balls, to deduce that the area of the boundary of the cylindrical set
have the same measure, this is equivalent to The conclusion then follows from the fact that The Pólya–Szegő inequality can be proved by combining the coarea formula, Hölder’s inequality and the classical isoperimetric inequality.
is smooth enough, the coarea formula can be used to write where
–dimensional Hausdorff measure on the Euclidean space
, we have by Hölder's inequality, Therefore, we have Since the set
is a ball that has the same measure as the set
, by the classical isoperimetric inequality, we have Moreover, recalling that the sublevel sets of the functions
is radial, one has and the conclusion follows by applying the coarea formula again.
, the Pólya–Szegő inequality can be proved by representing the Sobolev energy by the heat kernel.
is the heat kernel, defined for every