The interplay between two Euler­Lagrange operators relating to the nonlinear elliptic system

We establish the existence of multiple whirling solutions to a class of nonlinear elliptic systems in variational form subject to pointwise gradient constraint and pure Dirichlet type boundary conditions. A reduced system for certain SO ð n Þ -valued matrix ﬁelds, a description of its solutions via Lie exponentials, a structure theorem for multi-dimensional curl free vector ﬁelds and a remarkable explicit relation between two Euler–Lagrange operators of constrained and unconstrained types are the underlying tools and ideas in proving the main result.


Introduction
Let X & R n (with n ! 2) be a bounded domain with a C 1 boundary oX and consider the variational energy integral I½u; X ¼ Z X Fðx; u; ruÞ dx; ð1:1Þ where F ¼ Fðx; u; fÞ with ðx; u; fÞ 2 X Â R n Â R nÂn is a sufficiently regular realvalued Lagrangian satisfying certain bounds and growth at infinity and the competing maps u ¼ ðu 1 ; . . .; u n Þ are confined to the space of admissible incompressible Sobolev class maps A p u ðXÞ :¼ fu 2 W 1;p ðX; R n Þ : det ru ¼ 1 a:e: in X; u ¼ u on oXg for a fixed choice of exponent 1 p\1.
In the above formulation ru denotes the gradient matrix of u, an n Â n matrix field in X, that here is additionally required to satisfy the hard pointwise incompressibility constraint det ru ¼ 1 in X and u 2 C 1 ðoX; R n Þ is a pre-assigned boundary condition. The Euler-Lagrange system associated with the energy integral I½u; X over the space of admissible incompressible maps A p u ðXÞ then takes the form (cf., e.g., [1,3,4,6,19] where P ¼ PðxÞ is an unknown hydrostatic pressure field (a Lagrange multiplier) corresponding to the pointwise constraint det ru ¼ 1 and the differential operator L ¼ L½u; F takes the explicit form L½u; F ¼ 1 2 ½cof ru À1 fdiv F f ðx; u; ruÞ ½ À F u ðx; u; ruÞg: ð1:3Þ The divergence operator ''div'' in the first term on the right acts row-wise on the matrix field F f ðx; u; ruÞ whilst cof ru denotes the cofactor matrix of ru. Note that in view of det ru ¼ 1 the cofactor matrix cof ru is invertible: det cof ru ¼ ðdet ruÞ nÀ1 ¼ 1 and ½cof ru À1 ¼ ½ru t . Without going into technical details we recall that the system is formally the Euler-Lagrange equation associated with the unconstrained variational energy integral 1 I P uc ½u; X ¼ Z X F P uc ðx; u; ruÞ dx ¼ Z X fFðx; u; ruÞ À 2PðxÞðdet ru À 1Þg dx: ð1:4Þ (Notice that I P uc ½u; X ¼ I½u; X whenever u 2 A p u ðXÞ.) Here by a solution to the system (1.2) we mean a pair ðu; PÞ where u is of class C 2 ðX; R n Þ \ CðX; R n Þ, P is of class C 1 ðXÞ \ CðXÞ and the pair satisfy the system (1.2) in the pointwise (classical) sense. If the choice of P is clear from the context we often abbreviate by saying that u is a solution.
A good motivating source for considering such energies and classes of maps comes from the nonlinear theory of elasticity where the pair (1.1)-(1.2) describe a mathematical model of an incompressible hyperelastic material subject to pure displacement boundary conditions with the resulting extremisers-equivalently critical points or solutions to the associated Euler-Lagrange system-and minimisers serving as the equilibrium states and physically stable displacement fields. (For more on this see [1,3,4,6,18,19,21,24] and for other motivations see [2,9,10,12,14,16,17,20,23,26,27] and the references therein.) Whilst the methods of critical point theory provide a standard and efficient way of establishing the existence of (multiple) solutions to variational problems, due to the complex nature of the incompressibility constraint on the gradient of the competing maps, here, these methods drastically fail and are not applicable. In more technical terms the space A p u ðXÞ is far from being a Hilbert or Banach manifold whilst due to the a priori unknown regularity of the pressure field P, and integrability of the Jacobian determinant det ru, the unconstrained energy integral I P uc need not be everywhere well-defined, let alone, being continuously Frechet differentiable.
In this paper, we confine to Fðx; u; fÞ ¼ Fðr; juj 2 ; jfj 2 Þ with F ¼ Fðr; s; nÞ being a twice continuously differentiable Lagrangian satisfying suitable growth, coercivity and convexity properties (see below for more). Here r ¼ jxj, s ¼ juj 2 ¼ hu; ui denotes the 2-norm squared of u 2 R n and n ¼ jfj 2 ¼ Tr ff t fg ¼ Tr fff t g is the Hilbert-Schmidt norm squared of f 2 R nÂn . Thus with this notation in place we have ð1:5Þ where jruj 2 ¼ Trf½ru t ½rug ¼ Trf½ru½ru t g, whilst referring to the Euler-Lagrange differential operator L ¼ L½u; Further expansion then gives 2 2 The identity map u x is one solution to this system in view of the vector field L½u x ¼ r½F n À F s x with F n ¼ F n ðr; r 2 ; nÞ, F s ¼ F s ðr; r 2 ; nÞ being a gradient field in X.

ð1:8Þ
Our primary task is to establish the existence of multiple solutions to the system R½ðu; PÞ; X in (1.2) with L ¼ L½u as given in (1.7). We do so by way of analysing a reduced energy and an associated PDE system for certain SOðnÞ-valued matrix fields. The solutions u here are in the form of topologically whirling incompressible self-maps of the underlying spatial domain satisfying juj ¼ jxj and ujuj À1 ¼ Q½fðyÞxjxj À1 (see Sect. 2 for details) whose analytic and geometric features are intimately linked to those of the Lie group SOðnÞ and its Lie algebra of skew-symmetric matrices soðnÞ.  (see Sect. 4 for details). A thorough analysis then leads to a remarkable relationship between the two systems and their corresponding differential operators given by As a result the task of resolving the PDE L½u ¼ rP shifts to verifying whether and when the vector field on the right-hand side is a gradient. This analysis will be carried out by studying certain classes of curl free vector fields and an associated structure theorem paving the way for the main existence and multiplicity result formulated and proved in the final two sections of the paper. Assumptions on F. Let us end by describing the regularity and convexity assumptions imposed on the Lagrangian F ¼ Fðr; s; nÞ. First we assume throughout that F 2 C 2 ðUÞ where U ¼ ½a; bÂ0; 1½Â0; 1½& R 3 . Next we assume F to be bounded from below on U with F n [ 0, F nn ! 0. Moreover, we assume that for every compact set K &0; 1½ there exist real constants c 0 ; c 1 ; c 2 depending on K such that, for p [ 1, Finally, the twice continuously differentiable function f7 !Fðr; r 2 ; n þ r 2 f 2 Þ is assumed to be uniformly convex in f for all a r b and f 2 R.
Let X & R n (n ! 2) be a bounded domain and let u 2 W 1;1 ðX; R n Þ satisfy juj [ 0 a.e. in X. We decompose u into a radial part R u and a spherical part S u , respectively, by setting R u ¼ juj and S u ¼ ujuj À1 . As u is (weakly) differentiable basic calculation gives rR u ¼ juj À1 ½ru t u; rS u ¼ juj À1 I n À ujuj À1 ujuj À1 ru; ð2:1Þ where I n denotes the n Â n identity matrix and as before ½ru t is the transpose of ½ru. We also introduce a pair of matrix-fields associated with u and intertwined with the PDE: X½u :¼ ½ru t ½ru À I n ; Y½u :¼ ½ru½ru t À I n : ð2:2Þ These in a way measure the closeness of the gradient field ru to the orthogonal group OðnÞ and hence the deformation u to a rigid motion by Liouville's theorem (evidently ru 2 OðnÞ () X½u 0 () Y½u 0). Let us proceed by listing some of the main quantities associated with u in terms of its radial and spherical parts. Lemma 1 With the notation on R u ¼ juj and S u ¼ ujuj À1 as above the following identities hold: If, additionally, u is twice differentiable, then Proof These identities are all consequences of the definition and follow by direct differentiation upon noting jS u j 2 ¼ 1, ½rS u t S u ¼ 0. The details are left to the reader.
Proposition 1 The partial differential action L½u in (1.8) can be formulated in the radial and spherical parts R u and S u as: where the arguments of F and all subsequent derivatives are evaluated at the vector point ðjxj; juj 2 ; jruj 2 Þ ¼ ðr; ð2:7Þ Proof We invoke the definition of L½u in (1.8) and the identities gathered in Lemma 1. For the vector field B½u we note that rðjuj 2 Þ ¼ 2R u rR u . Premultiplying by ½ru t ½ru gives (2.5). The identity (2.6) then follows upon noting that rjxj ¼ x=jxj. The descriptions of the vector fields A½u, D½u and E½u are taken directly from identities (ii)-(vii) in Lemma 1.
In this section, we specialise to the situation where X & R n is a symmetric open bounded annulus, for definiteness, X ¼ X n ¼ X n ½a; b :¼ fx 2 R n : a\jxj\bg with b [ a [ 0 and u I X (identity map). The choice of u is to avoid unnecessary technicalities without losing too much generality, whilst the choice of domain geometry is prompted by applications to multiplicity results we have in mind for later on (compare with, e.g., [13,22] as well as [1,6,7,11,24,28]). For a self-map u 2 CðX n ; X n Þ, recall the notation introduced in the previous section, specifically, the radial part R u and the spherical part S u given by R u :¼ juj 2 CðX n ; ½a; bÞ and S u :¼ ujuj À1 2 CðX n ; S nÀ1 Þ, respectively. If u x on oX n then R u a and R u b on the inner and outer components of oX n , respectively, whilst S u h on oX n . Furthermore, due to the cartesian product structure of X n , the spherical part S u can be seen, with a slight abuse of notation, to verify S u 2 denotes the identity map of the unit sphere. Here we write CðS nÀ1 ; S nÀ1 ; deg ¼ dÞ (d 2 Z) for the connected component of the mapping space CðS nÀ1 ; S nÀ1 Þ consisting of maps with Hopf degree d. As a result S u represents an element of the fundamental group p 1 ½CðS nÀ1 ; S nÀ1 ; deg ¼ 1Þ ffi p 1 ½SOðnÞ (for more on this see [8,24,25,29,30]). Conversely any map S ¼ SðrÞ in Cð½a; b; CðS nÀ1 ; S nÀ1 ; deg ¼ 1ÞÞ satisfying SðaÞ ¼ SðbÞ ¼ I S nÀ1 gives rise to a self-map u 2 CðX n ; X n Þ with u x on oX n through the recipe R u ðxÞ ¼ f ðjxjÞ and S u S, i.e., u : ðr; hÞ7 !ðf ðrÞ; SðrÞ½hÞ. Here f 2 Cð½a; b; ½a; bÞ is any function satisfying f ðaÞ ¼ a and f ðbÞ ¼ b (e.g., f ðrÞ r). In what follows we look at particular classes of self-maps u whose spherical parts S u result from an SOðnÞ-valued matrix field Q as described in (a)-(b) below.
(a) Twists u 2 CðX n ; X n Þ. By a generalised twist or simply a twist we understand a self-map u whose radial and spherical parts are given by ð3:1Þ Here the curve Q 2 Cð½a; b; SOðnÞÞ is referred to as the twist path associated with u. To ensure u x on oX ¼ oX n we set QðaÞ ¼ QðbÞ ¼ I n where I n is the n Â n identity matrix. In this event the twist path is a closed curve in SOðnÞ based at I n thus representing an element of p 1 ½SOðnÞ ffi Z 2 (n ! 3) and ffi Z (n ¼ 2). Here we refer to Q ¼ QðrÞ as the twist loop associated to u. Now subject to the differentiability of the twist path Q (hereafter we write _ and so a direct calculation (see below) leads to det ru ¼ 1. If u is twice differentiable then by taking second derivatives it can be easily seen that DR u ¼ ðn À 1Þ=jxj Lemma 2 Let u be a twist associated with Q 2 Cð½a; b; SOðnÞÞ \ C 1 ða; b½; SOðnÞÞ: Then the following hold: If, moreover, the matrix field Q 2 Cð½a; b; SOðnÞÞ \ C 2 ða; b½; SOðnÞÞ then we have giving (v). Next (vi) results from taking the trace of either of X½u or Y½u. Now moving to the next part, for the Laplacian we use Du ¼ R u DS u þ 2rS u rR u þ DR u S u along with the earlier calculation of the constituting terms. The final identity can be pieced together using ingredients already gathered in the earlier part of the lemma.
Proposition 2 Let u be a twist with Q 2 Cð½a; b; SOðnÞÞ \ C 2 ða; b½; SOðnÞÞ: Then ð3:3Þ Here L is the differential operator in (1.8) and the arguments of F ¼ Fðr; s; nÞ and all subsequent derivatives are ðr; s; nÞ ¼ ðr; Proof We justify the statement using Proposition 1 and calculating the associated coefficients. First for A½u, noting jrR u j 2 ¼ 1, jrS u j 2 ¼ jxj À2 ½n À 1 þ j _ Qxj 2 , we have The remaining coefficients D½u and E½u can be taken from (iii) and (viii) in Lemma 2. The conclusion now follows upon noting (b) Whirls u 2 CðX n ; X n Þ. By a whirl map or a whirl for simplicity we understand a self-map u whose radial and spherical parts have the forms Here, we denote by y ¼ yðxÞ the vector of 2-plane radial variables ðy 1 ; . . .; y N Þ, defined, depending on the dimension n ! 2 being even or odd, as follows: 2j Þ 1=2 with 1 j N À 1 and y N ¼ x n . In the first case set d ¼ N and in the second case set d ¼ N À 1. It is now seen that for x 2 X n the vector y ¼ yðxÞ lies in the semi-annular domain where the three disjoint segments of oA n are defined as: C n ¼ oA n n ½ðoA n Þ a [ ðoA n Þ b (the flat part), ðoA n Þ a ¼ fy 2 oA n : kyk ¼ ag and ðoA n Þ b ¼ fy 2 oA n : kyk ¼ bg.
Note that x 2 ðoX n Þ a ¼ fjxj ¼ ag () yðxÞ 2 ðoA n Þ a , x 2 ðoX n Þ b ¼ fjxj ¼ bg () yðxÞ 2 ðoA n Þ b whilst the flat part C n does not correspond to any part of oX n .
With this notation in place let us now give a more explicit description of the SOðnÞ-valued matrix field Q ¼ Qðy 1 ; . . .; y N Þ defining the spherical part S u . 4 Let us denote by R½a the usual SOð2Þ matrix of rotation by angle a, specifically, : ð3:5Þ Here and below we write X7 !expfXg for the Lie exponential map of SOðnÞ, whose domain is the Lie algebra soðnÞ of n Â n skew-symmetric real matrices. Consideration of symmetry (see [16,17]  . . .; f d Þ. Now subject to a differentiability assumption on Q we can write (with o ' Q ¼ oQ=oy ' ) and again after direct but a little more involved calculation (see Lemma 3 below) it follows that det ru ¼ 1. Thus hereafter by a whirl we understand a self-map u as in (3.4) where the matrix field Q in S u has either form (3.6) or (3.7).
Lemma 3 Let u be a whirl associated with Q 2 CðA n ; SOðnÞÞ \ C 1 ðA n ; SOðnÞÞ: Then the following hold: h If, moreover, the matrix field Q 2 CðA n ; SOðnÞÞ \ C 2 ðA n ; SOðnÞÞ then we have Here o ' ; o 2 ' denote the first and second derivatives with respect to y ' ; whereas the gradients and Laplacians of the variables y ' ; y k are those with respect to x 1 ; . . .; x n .
Proof Recall that for a whirl we have R u ¼ jxj; S u ¼ QðyÞxjxj À1 . With rR u ¼ xjxj À1 identity (iii) follows at once from the corresponding identity in Lemma 1. For (i) referring to (3.8) we have ru ¼ R u rS u þ S u rR u and so as required. For X½u we use ½ru t ½ru ¼ rR u rR u þ R 2 u ½rS u t ½rS u together with

ð3:10Þ
For Y½u similarly we use ½ru½ru t ¼ R 2 u ½rS u ½rS u t þ R u ðrS u rR u S u þ S u rS u rR u Þ þ jrR u j 2 S u S u together with rS u rR u ¼ jxj À2 P ' hry ' ; xio ' Qx and hence which then gives Y½u. Note that here we have made use of the relation hry j ; ry k i ¼ d jk . Next, regarding jruj 2 we have upon recalling (v) in Lemma 1, jo ' Qxj 2 : Some details are straightforward and hence omitted. Here and below we use the observation that hQ t o ' Qx; have hp i ; q j i ¼ 0 for all 1 i; j N (as above), it follows from Lemma 3.1 in [16] that det½I n þ P N j¼1 p j q j ¼ 1 which then gives (ii). Turning to the Laplacian we recall DR u ¼ ðn À 1Þ=jxj and compute in a similar way As a result using Du ¼ R u DS u þ 2rS u rR u þ DR u S u we thus obtain A calculation using ingredients already gathered in the proof also verifies (viii).
Proposition 3 Let u be a whirl map with Q 2 CðA n ; SOðnÞÞ \ C 2 ðA n ; SOðnÞÞ: Then the action of the partial differential operator L on u can be described in terms of Q as ð3:13Þ The arguments of F ¼ Fðr; s; nÞ in (3.13) and all subsequent derivatives are ðr; s; nÞ ¼ ðjxj; juj 2 ; jruj 2 Þ ¼ ðr; r 2 ; n þ P N '¼1 jo ' Qxj 2 Þ: Proof We use Proposition 1 and similar to the argument in Proposition 2 proceed by computing the various coefficients associated with L½u as described by (2.4)-(2.7). Indeed for A½u we have where for the last term on the right we have Regarding the vector field B½u we have ho ' Qx; o k Qxihry k ; xiry ' ! :

ð3:16Þ
For C½u the calculation is again similar and we have ð3:17Þ Finally for D½u ¼ ½ru t Du and E½u ¼ À½ru t u we refer to (viii) and (iii) in Lemma 3. Putting these together and noting that rF n ¼ F nn rjruj 2 þ ð2F sn þ jxj À1 F rn Þx gives the desired conclusion.

Derivation of the relation between the constrained and unconstrained operators
We now consider restricting the energy integral I to the subclass of admissible whirls hence obtaining a restricted energy integral (called H below) for the angle of rotation vector function as in (3.6) Here J ¼ ð2pÞ d JðyÞ is the Jacobian for the change of variables from x ¼ ðx 1 ; . . .; x n Þ to y ¼ ðy 1 ; . . .; y N Þ with JðyÞ ¼ y 1 Á Á Á y d . In particular note that when n ¼ 2d is even JðyÞ ¼ y 1 . . .y N whereas when n ¼ 2d þ 1 is odd JðyÞ ¼ y 1 . . .y NÀ1 and so the last variable y N does not appear in this Jacobian product. One important implication here is that J is always strictly positive in A n . Moving forward we now aim to extremise H½f; A n over the space of admissible vector functions B p m ½A n , defined for m ¼ ðm 1 ; . . .; m d Þ 2 Z d and p ! 1, by B p m ½A n ¼ ff 2 W 1;p ðA n ; R d Þ : f 0 on ðoA n Þ a ; f 2mp on ðoA n Þ b g. It is not difficult to see that the resulting Euler-Lagrange system here takes the form (for 1 i d; m 2 Z d ) Proof Let us fix a solution f as described and set g ¼ f þ u, g 2 B p m ½A n . Then evidently Fðz; z 2 ; f 2 Þ À Fðz; y 2 a jru a j 2 JðyÞ dy: ð4:4Þ Here, we are using jrg a j 2 À jrf a j 2 ¼ jrðg a À f a Þj 2 þ 2rf a Á rðg a À f a Þ and the fact that f is a solution to (4.2) to deduce that the first and second integrals on the second line in (4.4) vanish. In particular the quantity on the left is non-negative giving the minimality of f in B p m ½A n . Now if f, g are both solutions as described in B p m ½A n then arguing as above H½g; A n ¼ H½f; A n . Thus the integral on the right in (4.4) vanishes and so in view of the strict inequalities F n ; J [ 0 in A n it follows at once that f g.

ð4:5Þ
Proof Starting from the expression on the left-hand side and expanding the divergence term on the left, a straightforward differentiation gives

ð4:6Þ
Here, we have made use of the relation Regarding the expressions on the right-hand side of (4.5) and by evaluating the individual terms it can be seen that (below 1 ' N and 1 k N) ð4:7Þ The third identity above along with the inner product relation hry ' ; 2 when 1 ' d (and zero when n is odd and ' ¼ N), whilst using the third and fourth identities we can obtain We note that the above expressions (specifically the sums on the left-hand side in the two identities above) form precisely the coefficients of F n in (4.5). In much the same way using the second and fourth identities in (4.7) lead to where use has been made of the orthogonality relations hw i ; ½w j ? i ¼ 0 (1 i; j N) along with h½w i ? ; ½w j ? i ¼ 0 (i 6 ¼ j) and h½w j ? ; ½w j ? i ¼ y 2 j . As a result we can write ð4:11Þ which forms the coefficient of 2F nn in (4.5). Substituting back the relevant terms and taking into account the cancellations gives the required relation.
This brings us to the following result bridging the differential operator action L½u for a whirl u associated with the matrix-field Q ¼ Q½f and the system RS½f; A n . Theorem 2 Let u be a whirl map associated with the matrix field Q ¼ Q½f where f ¼ ðf 1 ; . . .; f d Þ is of class C 2 ðA n ; R d Þ [see (3.6)-(3.7)]. Then the constrained PDE system R½ðu; P; XÞ and the unconstrained system RS½f; A n are directly related to one-another via the identity The arguments of F s ¼ F s ðr; s; nÞ and F n ¼ F n ðr; s; nÞ are ðr; s; nÞ ¼ ðjxj; juj 2 ; jruj 2 Þ and the coefficients A i ¼ A i ðy; rfÞ in (4.12) are exactly as given by (4.3).
Proof Starting with the description of L½u À rF n as in (3.13) a close inspection reveals ð4:13Þ Likewise moving to the next set of terms in (3.13) it is seen that hry ' ; xiQ t o ' Qx; ð4:14Þ and in a similar way ð4:15Þ Recalling (4.5) and noting that by skew-symmetry hQ t o ' Qx; o k Q t o k Qxi ¼ 0, (4.12) follows by putting the above fragments together and rearranging terms.
Corollary 1 Under the assumptions of the previous theorem suppose that the vector function f ¼ ðf 1 ; . . .; f d Þ solves the restricted system RS½f; A n : Then denoting by u ¼ Q½fðxÞ the whirl map associated with the matrix-field Q ¼ Q½f we have Proof As div½A i ðy; rfÞrf i ¼ 0 for each 1 i d the assertion follows from (4.12).

The operator
L½u and a gradient-curl analysis for 2-plane n-vector fields Returning to the Euler-Lagrange system R½ðu; PÞ; X we now aim to discuss the solvability of this system by taking advantage of the results of the previous section. Recall that if a whirl u associated with the SOðnÞ-valued matrix field Q ¼ Q½f is a solution to R½ðu; PÞ; X then the vector function f ¼ ðf 1 ; . . .; f d Þ is in turn a solution to the reduced system RS½f; A n . Hence in light of Corollary 1 we can write ð5:1Þ Proposition 5 Consider the vector field V ¼ VðxÞ of class C 1 ðX n ; R n Þ defined by Here y ¼ yðxÞ ¼ ðy 1 ; . . .; y N Þ 2 A n , x ¼ ðx 1 ; . . .; x n Þ 2 X n , w p ¼ w p ðxÞ, ½w q ? ¼ ½w q ðxÞ ? and a p ¼ a p ðyÞ, b q ¼ b q ðyÞ 2 C 1 ðA n Þ for all 1 p; q N: Then ð5:3Þ Here K q ¼ diagð0; . . .; 0; J; 0. . .; 0Þ; that is, the n Â n skew-symmetric block diagonal matrix with J its q th block. 5 Proof We verify this by direct evaluation of the curl. Indeed for 1 i\j n we have (with w p i denoting the i component of w p and ½w q i ? denoting the i component or ½w q ? ) ij and so the conclusion follows at once by substitution. Before discussing applications of the proposition to Theorem 2 let us pause briefly to take a closer look at some special cases of the statement and its implications.
• If a p (1 p N), b q (1 q d) are constant (in y) then curl V ¼ 2 P d q¼1 b q K q and so in particular curl V 0 () b q 0 for all q.
• If a p ðyÞ ¼ a p ðy p Þ; b q ðyÞ ¼ b q ðy q Þ for all 1 p; q N then with _ a p ¼ da p =dy p and _ b q ¼ db q =dy q we have ð5:5Þ In particular, if b q 0 (1 q d) then curl V 0. In fact choosing U p ¼ U p ðy p Þ so that dU p =dy p ¼ y p a p we have V ¼ r 2b q K q :

ð5:6Þ
If, additionally, a p ðyÞ ¼ aðrÞ, b q ðyÞ ¼ bðrÞ for all 1 p; q N then we have ð5:7Þ by virtue of x ¼ P N p¼1 w p and thus P N p¼1 P N k¼1 w p w k À w k w p Â Ã 0.
Let us now direct the above analysis towards the system R½ðu; PÞ; X and its whirl solutions. Indeed setting V ¼ L½u À rF n as in (5.1) and invoking Proposition 5 gives curl ðL½u À rF n Þ ¼ curl o k a p y k ðyÞ w p w k À w k w p Â Ã ¼ X 1 p\k N o k a p y k À o p a k y p w p w k À w k w p Â Ã ; ð5:8Þ with a p ¼ ÀF n jrf p j 2 À F s . 6 Now (5.8) 0 (recall the PDE L½u ¼ rP) results in the vanishing of all the coefficients of ½w p w k À w k w p , i.e., o k a p =y k À o p a k =y p 0. We will analyse the implications of these conditions further in the following sections.
6 The complete solvability of RS½f,A n and D H = D H ðrÞ To get a better view of Theorem 2, the system RS½f; A n in (4.2) and the curl analysis in the previous section we take a closer look at the case Fðr; s; nÞ ¼ Hðr; sÞn where H ¼ Hðr; sÞ is a strictly positive function of class C 2 . The system (4.2) here is linear and has the decoupled form (with 1 i d and no summation over i): Combining this latter expression with the earlier sum above, therefore, gives