Gruber P. Convex and Discrete Geometry

184 Convex Bodies
�
�� �2 2
grad f (x) − λf (x) � dx
Hence,
B
�
� � 2 2
= (grad gu) + g u�u dx
B
�
� 2 2 2 2
= g (grad u) + 2gu grad g · grad u + u (grad g)
B
B
− grad(g 2 u) · grad u � �
dx +
K
g 2 u ∂u
∂n dt
�
� 2 2 2 2
= g (grad u) + 2gu grad g · grad u + u (grad g)
− g 2 (grad u) 2 − 2gu grad g · grad u � dx
�
= u 2 (grad g) 2 dx ≥ 0.
B
λ ≤
� � �2dx grad f (x)
B
�
B
f (x) 2 dx
where equality holds if and only if grad g(x) = o, i.e. g = const or f = const u.The
proof of (7) is complete.
Proposition (7) and the rearrangement theorem finally imply the desired
inequality:
λ1(B) =
� � �2dx grad u(x)
B
�
u(x) 2dx B
D
≥
,
� � �
grad ur 2dx
(x)
D
�
ur (x) 2dx ⎧�
� �2dx ⎪⎨ grad h(x)
D
≥ inf �
⎪⎩
h(x) 2 ⎫
⎪⎬
: h ... = λ1(D). ⊓⊔
dx ⎪⎭
Remark. It can be shown that equality holds precisely for circular membranes. The
first principal frequency for circular membranes can be expressed in terms of Bessel
functions. For refinements, generalizations and related modern material we refer to
Kac [558], Bandle [64], Payne [786], Protter [818] and Talenti [987]. Many of the
pertinent results are related to the following question.
Problem 9.1. What information about a compact body B in E d can be obtained from
the sequence of eigenvalues of the eigenvalue problem (5)?
D