157
(Q ) and, suppose (r l' r2,.··r n) the n solutions of the problems (Pi), (the existence of these
solutions in the regularity conditions (1,1) is classical), then the gradpi, Curlrj and Curlrk sets
constitute an orthogonal base of L 2 (Q)2 and of V.
(P4)
(f) The Curlq j and Curlrk are solutions of the problem:
-~u = Curi (curlu) = AU
divu = 0 into Q
u.n = 0 . curiu = 0 on y
and reciprocally every solution u of (P4) is solution of (PI) with u = Curlqj' qj solution of
(P3) if A"* 0 or a linear combination of the Curirk if A = O.
Before we give the demonstration of the theorem, we include some useful results to
the demonstration. We will note that the operator associated to the problem (PI) is selfadjoint but is not necessarily at compact resolvant. As a matter of fact, the boundary
conditions of the problem (P I) are not completely standard and the test space functions of the
variational problem associated with (P1) (which is here the space V), is not in all the cases
included into H 1( Ql The results that we give hereafter precisely state at which conditions
V is included into HI(Q)2, (which proves the existence of the L 2( Q )2 base associated with
(PI) as the operator is then at compact resolvant), and how the elements of V can be distorted
into the sum of a gradient and of a curl of scalar functions.
II. 2.2 Variational formulation and working space.
We first remind the definitions of the H (div) and H (curl) spaces and then the trace
theorems on these spaces and the Green's formulae necessary to establish the variational form
of the problem (PI). These results are developped in Giraut-Raviart (1986).
(11.2,4)
On this space u.n has a meaning in W 112 (y) and the Green's formula is valid for each
uE H(div) and for each p E H I( Q) :
(11.2,5)
(u, gradp) + (p, divu) = < u.n. p >
where <. , .> means the duality W 1/2 (y), H 1/2 (y) .
(II. 2,6)
H(curl) = {u E L 2 (Q)2, curlu E L2 (Q)}
On this space a (u).n has a meaning in H- 1I2 (y) and the Green's formula is valid for each
uE H(curi) and q E Hl( Q) :