Partial Differential Equations - Modelling and ... - ResearchGate

264 Y. Achdou
The spaces H s (R) are Hilbert spaces, with the inner product and norm:
∫
√
(w 1 ,w 2 ) H s (R) = (1 + ξ 2 ) s ŵ 1 (ξ)ŵ 2 (ξ)dξ, ‖w‖ H s (R) = (w, w) H s (R).
R
We refer to [Ada75] for the properties of the spaces H s (R). If s is a nonnegative
integer, we define the semi-norm
( s∑ ∥ |v| Hs (R) =
d l v ∥∥∥
2
∥dy l l=1
If s>0 is not an integer, we define |v| Hs (R) by
m ∥
|v| 2 H s (R) = ∑
d l v ∥∥∥
2
∥dy l
l=1
where m is the integer part of s.
L 2 (R)
∫ ∫
+
R R
L 2 (R)
) 1
2
.
(
) 2
d m v
dy
(y) − dm v
m dy
(z) m
|y − z| 1+2s ,
Some Weighted Sobolev Spaces on R +
Let L 2 (R + ) be the Hilbert space of square integrable functions on R + ,endowed
with the norm ‖v‖ L2 (R +) = ( ∫ R +
v(x) 2 dx) 1 2 and the inner product
(v, w) L 2 (R +) = ∫ R +
v(x)w(x)dx. LetV 1 be the weighted Sobolev space
{
V 1 = v ∈ L 2 (R + ), x ∂v }
∂x ∈ L2 (R + ) , (19)
which is a Hilbert space with the norm
(
∥ ∥∥∥
‖v‖ V 1 = ‖v‖ 2 L 2 (R + +) x ∂v
∂x∥
2
L 2 (R +)
) 1
2
. (20)
It is proved in [AP05a] that D(R + ) is a dense subspace of V 1 , and that the
following Poincaré inequality is true: for all v ∈ V 1 ,
‖v‖ L2 (R +) ≤ 2
∥ x dv
∥ . (21)
dx
∥
L2 (R +)
Thus the semi-norm |·| V 1: |v| V 1 = ‖x dv
dx ‖ L 2 (R +) is a norm equivalent to ‖·‖ V 1.
For a function v defined on R + , call ṽ the function defined on R by
( y
)
ṽ(y) =v(exp(y)) exp . (22)
2
By using the change of variable y =log(x), it can be seen that the mapping
v ↦→ ṽ is a topological isomorphism from L 2 (R + )ontoL 2 (R), and from V 1
onto H 1 (R). This leads to defining the space V s ,fors ∈ R, by: