156
11.2. Property of the special basis.
We note :
(11.2,1)
(11.2,2)
v = { U E L 2 (n) 2 , divu E L 2 (0) , curlu E L 2 (n) ; u . n = 0 on y}
V 0 (div 0, curiO) = { u E L 2 (n) 2, divu = 0; curlu = 0; U • n = 0 on y}
(11.2,3)
1.1, (.) the norm and the scalar product into L 2 (n) or L 2(0) 2.
and we consider the problems :
(PI)
-Au = AU into n ; u. n = 0 ; curlu = 0 on y
where u is a function of n into R 2
(P2)
-Ap = AP into n ; gradp. n = 0 on y
(P3)
-Aq = Aq into n ; q = 0 on y
and the n problems:
(Pi)
-dr = 0 into n ; r = 1 on y i ; r = 0 on Y j , j :;t i
with i = 1, ... ,n; j = O, .... ,n
where p, q, r are functions of n into R.
Some properties of the (PI) solutions are summarized in the
II. 2. 1 Theorem
(a) If (A., p) is a solution of (P2), then (A, gradp) is solution of (PI).
(b) If (11, q) is solution of (P3), then (11, Curlq) is solution of (PI). This property
particularly demonstrates the existence of solutions of (PI) with free divergence.
(c) If ri is solution of (Pi) then (0, CurIri) is solution of (PI).
(d) If n is simply connected then 0 is not an eigen value of (PI), otherwise the eigenspace associated to the 0 eigen value is the HO(div 0, curiO) space, of dimension n, produced
by the n solutions of the (Pi) problems.
(e) Suppose {Pi; i E N} a set of solutions of (P2) constituting an orthogonal base of
L 2(n) , suppose (qj ; j E N) a set of solutions of (P3) constituting an orthogonal base of L 2