Gruber P. Convex and Discrete Geometry

Fig. 9.3. Rearrangement of functions
9 Symmetrization 179
The following rearrangement theorem is indispensable for certain isoperimetric
inequalities of mathematical physics, see the next section. While this result was
known and used at least since Faber [316] and Krahn [615], a complete proof requires
tools of more recent geometric measure theory. See Talenti [987] for its history and
references to proofs. Compare also Bandle [64]. Here no proof is presented.
Theorem 9.7. Let f : Ed →[0, +∞) be locally Lipschitz and such that f (x) → 0
as �x� →∞. Then f r : Ed →[0, +∞) is also locally Lipschitz, f r (x) → 0 as
�x� →∞, and
�
�
� �2dx � � r 2dx.
grad f (x) ≥ grad f (x)
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(Since f and f r are locally Lipschitz, (grad f ) 2 and (grad f r ) 2 are measurable and
exist almost everywhere, see [696].)
9.4 Torsional Rigidity and Minimum Principal Frequency
We begin with the following quotation from Bandle [64], preface.
The study of ‘isoperimetric inequalities’ in a broader sense began with the conjecture
of St Venant in 1856. Investigating the torsion of elastic prisms, he observed
that of all cross-sections of given area the circle has the maximal torsional rigidity.
This conjecture was proved by Pólya in 1948. Lord Rayleigh conjectured that of
all membranes with given area, the circle has the smallest principal frequency. This
statement was proved independently by Faber and Krahn around 1923.
There is a large body of isoperimetric type inequalities in mathematical physics
dealing with eigenvalues of partial differential equations. In physical terms these
inequalities concern principal frequencies of membranes, torsional rigidity, bending
of beams, electrostatic capacity, etc. Major contributors are Saint-Venant, Faber,
Krahn, Pólya and Szegö, Osserman, Hersch, Payne, Talenti and others.
The above rearrangement theorem will be applied to give the best upper, respectively,
lower, bound for the torsional rigidity of an elastic cylindrical rod and for the
first principal frequency of a clamped membrane.
For more information, see the surveys and books of Pólya and Szegö [811],
Bandle [64], Mossino [757], Hersch [498], Payne [786] and Talenti [987].
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