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Functional Spaces 91
(5.7.5)
u
1 − x2
L 2 w (I)
Now, we introduce another space, namely
≤ K u ′ L 2 w (I), ∀u ∈ H 1 w(I) with u(±1) = 0.
(5.7.6) H 1
0,w(I) := u ∈ H 1
w(I) u(±1) = 0 .
The following norm will be adopted in H 1 0,w(I):
(5.7.7) u H 1 0,w (I) := u ′ L 2 w (I), ∀u ∈ H 1 0,w(I).
To show that this is a norm we have to check (2.1.3). Actually, by (5.7.5), the condition
u H 1 0,w (I) = 0 implies u ≡ 0. In practice, in H 1 0,w(I), it is sufficient to bound the
derivative of u in order to control the function itself.
As in the previous section, we would like to define Sobolev spaces H s w(I), where
s ≥ 0 is a real number. There are several approaches. A first approach consists in
constructing a suitable prolongation of the weight w and functions in L 2 w(I) to the whole
domain R. Then we can provide a definition involving the transformation F following
the guideline of section 5.6. Another definition can be based on a formula similar to
(5.6.8). However, the common practice is to define H s w(I) as an intermediate space
between H [s]
w (I) and H [s]+1
w (I), via a technique called interpolation (see for instance
lions and magenes(1972)). For the sake of simplicity, we shall not enter into the
details of this procedure.
The case w ≡ 1 has been extensively investigated by many authors. For a general
weight function, we cannot expect the various definitions to match. The literature on
this subject does not cover all the possible cases and the theory is often quite compli-
cated. Results have recently been published in maday(1990) on the interpolation of
Sobolev spaces with the Chebyshev weight function. Though the use of spaces obtained
by interpolation seems to be a promising approach in view of the applications, with
a few exceptions, one can easily end up with abstract intermediate spaces, without a
satisfactory characterization of their elements.