Page 19 - Textos de Matemática Vol. 34

REMARKS ON SOBOLEV EMBEDDINGS 9
n = 1 and 1 < p < ∞ there is a relationship between the approximation numbers 0
of E0(I) : W1p(Ω) → Lp(Ω) and certain eigenvalues, namely that (2.12) ak(E0) = μ−1/p (k ∈ N),
k
where μk is the kth eigenvalue of the one-dimensional p−Laplacian Dirichlet prob-
lem, namely
(2.13) (|u′|p−2 u′)′ + μ |u|p−2 u = 0 on I = (a, b), u(a) = u(b) = 0.
In fact, problem (2.13) arises when we seek to minimise the Rayleigh quotient R(u) = ∥u′ | Lp(I)∥
∥u | Lp(I)∥
over W1p(I)\{0} : the minimum μ1 is given by the smallest eigenvalue of (2.13). It
has been shown by Oˆtani [19] that the eigenvalues of (2.13) are given by
kπp p
μk =(p−1) b−a (k∈N)
and from this (2.12) follows immediately.
3. Problems
Here I give a sample of the more obvious problems which arise naturally from the preceding discussion.
(i) Can exact formulae, or at least genuine asymptotic expansions, be obtained
0
for the approximation numbers of the embedding of W1p(I) in Lq(I) for suitably
related p and q? The same question arises for weighted spaces and for various embeddings of higher-order Sobolev spaces on I.
(ii) Can the exact behaviour of the entropy numbers ek(E0) be determined for the 0
embedding E0 : W1p(I) → Lp(I)? The conventional wisdom is that entropy num- bers, unlike approximation numbers, are nasty nonlinear objects which are quite likely to oscillate considerably, and so it is unlikely that anything as sharp as the results of Theorem 2.2 could be obtained, and also improbable that limk→∞ ek(E0) exists. Nevertheless, it would be interesting to settle the matter one way or the other.
(iii) Can results like Theorem 2.2 be obtained for higher-dimensional domains, for example when I is replaced by a cube or a ball? An obstacle is that knowledge of the p−Laplacian in space dimensions greater than 1 is still far from complete. In particular, there seems to be no higher-dimensional analogue of Oˆtani’s results.
0