Page 12 - Textos de Matemática Vol. 34

2 DAVID E. EDMUNDS
or not limk→∞ ks/nak(id) exists. I shall report on some first steps which have been taken towards the resolution of this question, and shall also mention related problems which at the moment appear to be open.
2. Preliminaries
Let X and Y be Banach spaces and let T : X → Y be bounded and linear.
Given any m ∈ N, the mth approximation number am(T) of T is defined by am(T)=inf ∥T−F∥,
where the infimum is taken over all bounded linear maps F : X → Y with rank less than m. The mth entropy number em(T) of T is given by
e (T)=infε>0:T(U )⊂∪2m−1(b +εU )forsomeb,...,bm−1 ∈Y, m Xj=1jY12
where UX , UY are the closed unit balls in X, Y respectively. Details of the basic properties of these numbers will be found in [4], [6] and [10]. We write
α(T) = lim am(T) and β(T) = lim em(T). m→∞ m→∞
We shall principally be interested in the case in which Y = Lp(Ω), where 1 < p < ∞ and Ω is a bounded domain in Rn. For this case it turns out that (see [6], Chapter V) α(T) = β(T), and this common value is called the measure of non-compactness of T since T is compact if, and only if, β(T) = 0.
Now let Ω be a bounded domain in Rn, let 1 < p < ∞ and let Wp1(Ω) be the Sobolev space consisting of those functions u in Lp(Ω) all of whose first-order distributional derivatives Dju are also in Lp(Ω). This becomes a Banach space when endowed with the norm
1 p n p 1/p
underlying space domain is one-dimensional, that is, when n = 1, and we shall 0
always suppose that a,b ∈ R with a < b, and shall write I = (a,b). By W1p(I) we shall mean the completion of C0∞(I), the space of all smooth functions with compact support in I, with respect to the norm
equivalent to · | Wp1(I) on this subspace. Moreover, the elements of W1p(I) are absolutely continuous on [a, b] and vanish at the endpoints a and b.
Ω
|u| + j=1 |Dju| dx
u | Wp (Ω) :=
For most of the paper we shall be concerned with the special case in which the
01 ′ f|W(I) :=∥f|L(I)∥;
(2.1)
evidently W1p(I) is a closed subspace of Wp1(I) and the norm defined by (2.1) is
0
p p 0
.