Untitled - Cdm.unimo.it

An Example in Two Dimensions 283
Figure 13.2.1 - The set ℵ8 ⊂ ¯ Ω for α = β = − 1
2 .
Thus, we are concerned with finding a polynomial pn ∈ P ⋆,0
n such that
(13.2.2) −∆pn(η (n)
i ,η (n)
j ) = f(η (n)
i ,η (n)
j ), 1 ≤ i ≤ n − 1, 1 ≤ j ≤ n − 1.
The above set of equations is equivalent to a (n − 1) 2 × (n − 1) 2 linear system. We
examine for simplicity the case n = 4, although the generalization to other values of n
is straightforward. Let us define the vectors
¯pn ≡ pn(η (n)
1 ,η (n)
1 ),pn(η (n)
2 ,η (n)
1 ),pn(η (n)
3 ,η (n)
1 ),pn(η (n)
1 ,η (n)
2 ),pn(η (n)
2 ,η (n)
2 ),
¯fn ≡
pn(η (n)
3 ,η (n)
2 ),pn(η (n)
1 ,η (n)
3 ),pn(η (n)
2 ,η (n)
3 ),pn(η (n)
3 ,η (n)
3 )
f(η (n)
1 ,η (n)
1 ),f(η (n)
2 ,η (n)
1 ),f(η (n)
3 ,η (n)
1 ),f(η (n)
1 ,η (n)
2 ),f(η (n)
2 ,η (n)
2 ),
f(η (n)
3 ,η (n)
2 ),f(η (n)
1 ,η (n)
3 ),f(η (n)
2 ,η (n)
3 ),f(η (n)
3 ,η (n)
3 )
.
,