The aim of this section is to introduce and examine spherical whirls as potential extremisers of the energy integral \(\mathbb{F}\), that is, as solutions to the nonlinear system of Euler-Lagrange equations (1.6).
To this end, recall first the 2-plane radial variables \(\rho=(\rho _{1}, \dots, \rho_{N})\) from the previous section, defined as functions of the spatial variable \(x=(x_{1}, \dots, x_{n})\) on \(\mathbb{X}^{n}\) for n even and odd by [a] (n even) \(\rho_{j} = (x_{2j-1}^{2} + x_{2j}^{2})^{1/2}\) for \(1\leq j \leq k=n/2\) and likewise [b] (n odd) \(\rho_{j} = (x_{2j-1}^{2} + x_{2j}^{2})^{1/2}\) for \(1\leq j \leq k=(n-1)/2\) and \(\rho _{N}=x_{n}\) (i.e., \(j=(n+1)/2=N\)), respectively. Note that, as indicated earlier, here in order to simplify notation, we write \(N=N(n)\) as \(N=k\) when \(n=2k\) and \(N=k+1\) when \(n=2k+1\). Now \(\rho= (\rho_{1},\dots,\rho_{N})\) lies in the semi-annulus \({\mathbb {A}}_{N} \subset {\mathbb {R}}^{N}\) given by (a) (n even) \({\mathbb {A}}_{N} = \{ \rho\in {\mathbb {R}}^{k}_{+} : a<\|\rho\|<b\}\) with \(n=2k\), and (b) (n odd) \(\{ \rho\in {\mathbb {R}}^{k}_{+} \times {\mathbb {R}}: a <\|\rho\|<b \} \) with \(n=2k+1\), respectively. We write \((\partial {\mathbb {A}}_{N})_{a} = \{ \rho\in\partial {\mathbb {A}}_{N}:\|\rho\|= a\}\), \((\partial {\mathbb {A}}_{N})_{b} = \{ \rho\in\partial {\mathbb {A}}_{N}: \|\rho\|=b\}\) and \(\Gamma_{N} = \partial {\mathbb {A}}_{N} \setminus\{\rho\in\partial {\mathbb {A}}_{N}: \|\rho\|=a \text{ or } \|\rho\|=b\}\) to denote the three components of the boundary \(\partial {\mathbb {A}}_{N}\). Note that \(x=(x_{1}, \dots, x_{n}) \in \partial \mathbb{X}_{a}^{n} \iff\rho=(\rho_{1}, \dots,\rho_{N}) \in(\partial {\mathbb {A}}_{N})_{a}\), and likewise \(x \in\partial \mathbb{X}_{b}^{n} \iff\rho\in(\partial {\mathbb {A}}_{N})_{b}\).
With this notation in place, we now define a spherical whirl as a map \(u \in\mathscr{C}(\overline{\mathbb{X}}^{n}, {\mathbb {S}}^{n-1})\) having the form
$$ u: x \mapsto {\mathbf{Q}}(\rho) \theta= {\mathbf{Q}}(\rho_{1}, \dots, \rho_{N}) x|x|^{-1}, \quad x \in\overline{\mathbb{X}}^{n}, $$
(2.1)
where \(\rho=(\rho_{1}, \dots, \rho_{N})(x)\) is the vector of 2-plane variables as described above and \({\mathbf{Q}}\in\mathscr{C}(\overline{{\mathbb {A}}}_{N} ,\mathbf{SO}(n))\). Later on, especially in studying the extremising properties of spherical whirls, we may need to improve the regularity of u to \(\mathscr{C}^{2}\), but for the sake of this general definition, continuity is enough.
Generally we think of a \(u \in\mathscr{C}(\overline{\mathbb{X}}^{n}, \mathbb {S}^{n-1})\) as being rotationally symmetric iff it is invariant under all rotations R, that is, iff it satisfies \(u(x) = R u(R^{t} x)\) for all \(x \in \mathbb{X}^{n}\) and \(R \in\mathbf{SO}(n)\). For the sake of this paper, however, we think of weakening this condition and referring to u as being symmetric iff u is invariant under all rotations \(R \in\mathbb{T} \subset\mathbf{SO}(n)\), that is, \(u(x) = R u(R^{t} x)\) for all \(x \in \mathbb{X}^{n}\) and \(R \in\mathbb{T}\) where \(\mathbb {T}\) is a fixed maximal torus in \(\mathbf{SO}(n)\), that is, a maximal commutative subgroup in \(\mathbf{SO}(n)\). Now we demand any spherical whirl u to be invariant under the subgroup \(\mathbb{T} \subset\mathbf{SO}(n)\) of all planar rotations in the \((x_{2j-1},x_{2j})\)-planes with j ranging as described above. It is well known that here \(\mathbb{T}\) is a maximal torus in \(\mathbf{SO}(n)\) and as such is maximally commutative. This therefore fixes the range of Q and gives \({\mathbf{Q}}\in\mathscr {C}(\overline{{\mathbb {A}}}_{N}, \mathbb{T} )\), since if u is invariant under \(\mathbb{T}\), then
$$\begin{aligned}[b] R u \bigl(R^{t} x\bigr) &= R {\mathbf{Q}}(\rho_{1}, \ldots, \rho_{N}) R^{t} x \big|R^{t} x\big|^{-1} \\ &= R {\mathbf{Q}}(\rho_{1}, \ldots, \rho_{N}) R^{t} x |x|^{-1} \\ &= {\mathbf{Q}}(\rho_{1}, \ldots, \rho_{N}) x |x|^{-1} = u(x), \quad\forall x \in \overline{\mathbb{X}}^{n}, \forall\rho\in\overline{{\mathbb {A}}}_{N}, \forall R \in \mathbb{T},\end{aligned} $$
(2.2)
and so for each \(\rho\in\overline{{\mathbb {A}}}_{N}\), \({\mathbf{Q}}(\rho)\) commutes with \(\mathbb{T}\), which by definition of \(\mathbb{T}\) being maximal commutative implies that \({\mathbf{Q}}(\rho) \in\mathbb{T}\). Note that in the above we have taken advantage of the fact that \(\rho(Rx)=\rho\) for all \(x \in \mathbb{X}^{n}\), \(R \in\mathbb{T}\). In conclusion, for the outlined reasons of commutativity and symmetry, the spherical whirls must take the form
$$ u(x) = {\mathbf{Q}}(\rho_{1}, \ldots, \rho_{N}) x|x|^{-1}, \quad\rho=\rho (x)=(\rho_{1}, \ldots, \rho_{N}) \in\overline{{\mathbb {A}}}_{N}, x \in\overline{\mathbb{X}}^{n}, $$
where the mapping \({\mathbf{Q}}={\mathbf{Q}}(\rho_{1}, \ldots, \rho_{N})\) admits the specific block diagonal matrix form
$$ {\mathbf{Q}}(\rho_{1}, \dots, \rho_{N}) = \textstyle\begin{cases} \operatorname{diag}(\mathcal{R}[f_{1}] ,\dots, \mathcal{R}[f_{k}]) &\text{for } n=2k,\\ \operatorname{diag}( \mathcal{R}[f_{1}],\dots, \mathcal{R}[f_{k}] , 1) &\text{for } n=2k+1. \end{cases} $$
(2.3)
Here, for \(1\leq l \leq k\), \(f_{l} \in\mathscr{C}(\overline{{\mathbb {A}}}_{N}, {\mathbb {R}})\) satisfy \(f_{l} \equiv0\) on \((\partial {\mathbb {A}}_{N})_{a}\) and \(f_{l} = 2\pi m_{l} + z_{l}\) on \((\partial {\mathbb {A}}_{N})_{b}\). The latter will ensure in view of (2.1)-(2.3) that \(u=\varphi\) on \(\partial \mathbb{X}^{n}\). We start by calculating some of the quantities associated with spherical whirls.
Lemma 2.1
For a spherical whirl
\(u = {\mathbf{Q}}(\rho_{1}, \dots, \rho_{N}) x|x|^{-1}\)
with
\(x \in\overline{\mathbb{X}}^{n}\)
and
\((\rho_{1}, \ldots, \rho_{N}) \in \overline{{\mathbb {A}}}_{N}\)
and subject to
\({\mathbf{Q}}\in\mathscr{C}(\overline{{\mathbb {A}}}_{N}, \mathbf{SO}(n)) \cap \mathscr{C}^{1}({{\mathbb {A}}}_{N}, \mathbf{SO}(n))\), we have
-
\(\displaystyle\nabla u = \nabla\bigl({\mathbf{Q}}(\rho_{1}, \ldots, \rho_{N}) x|x|^{-1}\bigr)= \frac{{\mathbf{Q}}(\mathbf{I}_{n} - \theta\otimes\theta )}{r} + \sum_{l=1}^{N} {\mathbf{Q}}_{,l} \theta\otimes\nabla\rho_{l}\),
-
\(\displaystyle|\nabla u|^{2} = \operatorname{tr} \bigl\{ [\nabla u] [\nabla u]^{t} \bigr\} = \frac{n-1}{r^{2}} + \sum_{l=1}^{N} |{\mathbf{Q}}_{,l} \theta|^{2}\).
If additionally
\({\mathbf{Q}}\in\mathscr{C}^{2}({{\mathbb {A}}}_{N}, \mathbf{SO}(n))\), that is, Q
is twice continuously differentiable on
\({{\mathbb {A}}}_{N}\), then
-
\(\displaystyle\Delta u = \sum_{l=1}^{N} \biggl[ {\mathbf{Q}}_{,ll}\theta+ \frac{2}{r} {\mathbf{Q}}_{,l} \nabla\rho_{l} + {\mathbf{Q}}_{,l} \theta ( \Delta\rho_{l} - \frac{2 \rho_{l}}{r^{2}} ) \biggr] - \frac{n-1}{r^{2}}{\mathbf{Q}}\theta\).
Here
\({\mathbf{Q}}_{,l}\)
and
\({\mathbf{Q}}_{,ll}\)
denote the first- and second-order derivatives of
Q
with respect to
\(\rho_{l}\)
respectively, whereas
\(\nabla\rho_{l}\)
and
\(\Delta\rho_{l}\)
denote the gradient and Laplacian of
\(\rho_{l}\)
with respect to the spatial variable
\(x=(x_{1}, \ldots, x_{n})\).
Proof
Firstly a straightforward differentiation using the given formulation of the map \(u={\mathbf{Q}}(\rho_{1}, \ldots, \rho_{N}) x|x|^{-1}\) gives
$$\begin{aligned}[b] \nabla u &= {\mathbf{Q}}\nabla\theta+ \sum_{l=1}^{N} {\mathbf{Q}}_{,l} \theta\otimes \nabla\rho_{l} \\ &= \frac{1}{r} ( {\mathbf{Q}}- {\mathbf{Q}}\theta\otimes\theta ) + \sum _{l=1}^{N} {\mathbf{Q}}_{,l} \theta\otimes\nabla \rho_{l},\end{aligned} $$
(2.4)
where \(r = |x| = \sqrt{\rho_{1}^{2} + \cdots+ \rho_{N}^{2}}\). With the aid of this we can then calculate the Hilbert-Schmidt norm of the gradient ∇u by writing
$$\begin{aligned} |\nabla u|^{2} &= \operatorname{tr} \bigl\{ [ \nabla u] [\nabla u]^{t} \bigr\} \\ &= \operatorname{tr} \Biggl\{ \frac{1}{r^{2}}({\mathbf{I}}_{n} - {\mathbf{Q}}\theta\otimes {\mathbf{Q}}\theta) + \frac{1}{r} \sum_{l=1}^{N}({\mathbf{Q}}- {\mathbf{Q}}\theta\otimes\theta) (\nabla \rho_{l} \otimes {\mathbf{Q}}_{,l} \theta) + \sum_{l=1}^{N} {\mathbf{Q}}_{,l} \theta \otimes {\mathbf{Q}}_{,l} \theta \Biggr\} \\ &= \frac{n-1}{r^{2}} + \frac{1}{r}\sum_{l=1}^{N} \bigl\{ \langle {\mathbf{Q}}\nabla\rho_{l} , {\mathbf{Q}}_{,l} \theta\rangle- \langle {\mathbf{Q}}\theta, {\mathbf{Q}}_{,l} \theta\rangle \langle\theta,\nabla \rho_{l} \rangle + r|{\mathbf{Q}}_{,l} \theta|^{2} \bigr\} \\ &= \frac{n-1}{r^{2}} + \frac{1}{r}\sum_{l=1}^{N} \bigl\{ \langle {\mathbf{Q}}\nabla\rho_{l} , {\mathbf{Q}}_{,l} \theta\rangle + r|{\mathbf{Q}}_{,l} \theta|^{2} \bigr\} , \end{aligned}$$
(2.5)
where in deriving the last line we used the fact that \(({\mathbf{Q}}^{t} {\mathbf{Q}})_{,l}=0\) implying in turn that the matrix product \({\mathbf{Q}}^{t} {\mathbf{Q}}_{,l}\) is skew-symmetric and subsequently \(\langle {\mathbf{Q}}\theta, {\mathbf{Q}}_{,l} \theta\rangle=0\). We now move on to the inner product term \(\langle {\mathbf{Q}}\nabla\rho_{l}, {\mathbf{Q}}_{,l} \theta \rangle\). First, upon recalling that Q is of the form (2.3), a straightforward differentiation gives
$$ {\mathbf{Q}}^{t} {\mathbf{Q}}_{,l} = \textstyle\begin{cases} \operatorname{diag}(\partial_{l} f_{1}{\mathbf{J}}, \dots, \partial_{l} f_{k} {\mathbf{J}}) &\text{if } n=2k,\\ \operatorname{diag}(\partial_{l} f_{1}{\mathbf{J}}, \dots,\partial_{l} f_{k} {\mathbf{J}},0) &\text{if } n=2k+1, \end{cases} $$
(2.6)
where recalling and referring to (2.3) and (2.6), J and \(\mathcal{R}[f]=\operatorname{exp}(f {\mathbf{J}})\) are the \(2 \times2\) matrices
$$ {\mathbf{J}}= \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \quad\mbox{and} \quad \mathcal{R}[f] = \begin{bmatrix} \cos f & - \sin f \\ \sin f & \quad\cos f \end{bmatrix} , $$
(2.7)
respectively. If we write \(y_{l} = (x_{2l-1} ,x_{2l})\) for \(1 \leq l \leq k\) if \(n = 2k\) and additionally \(y_{2k+1} =x_{2k +1}\) if \(n=2k+1\), then we have
$$ {\mathbf{Q}}^{t} {\mathbf{Q}}_{,l} \theta= \textstyle\begin{cases} \frac{1}{|x|} (\partial_{l} f_{1} {\mathbf{J}}y_{1} , \dots, \partial_{l} f_{k} {\mathbf{J}}y_{k}) &\text{if } n=2k, \\ \frac{1}{|x|} (\partial_{l} f_{1} {\mathbf{J}}y_{1} , \dots, \partial_{l} f_{k} {\mathbf{J}}y_{k} , 0) &\text{if } n=2k + 1. \end{cases} $$
(2.8)
Furthermore, differentiating \(\rho_{j}\) and using identities (1.9) and (1.10), we see that
$$ \nabla\rho_{j} =\frac{1}{\rho_{j}} (0, \dots,y_{j}, \dots,0), \quad1 \le j \le N. $$
(2.9)
Therefore, by substitution, the following inner product identity is seen to hold (note that here there is no summation over \(1 \le j, l \le N\) and the penultimate equality excludes the relatively simpler case \(j=N\) for n odd in which the identity trivially holds):
$$ \langle {\mathbf{Q}}\nabla\rho_{j}, {\mathbf{Q}}_{,l} \theta\rangle= \bigl\langle \nabla \rho_{j}, {\mathbf{Q}}^{t} {\mathbf{Q}}_{,l} \theta \bigr\rangle = \frac{\partial_{l} f_{j}}{|x| \rho_{j}} \langle y_{j} , {\mathbf{J}}y_{j} \rangle= 0. $$
(2.10)
It now follows, upon referring to (2.5), that
$$ |\nabla u|^{2} = \frac{n-1}{r^{2}} + \frac{1}{r}\sum _{l=1}^{N} \bigl\{ \langle {\mathbf{Q}}\nabla\rho_{l} , {\mathbf{Q}}_{,l} \theta\rangle + r|{\mathbf{Q}}_{,l} \theta|^{2} \bigr\} = \frac{n-1}{r^{2}} + \sum_{l=1}^{N} |{\mathbf{Q}}_{,l} \theta|^{2} . $$
(2.11)
Finally, the Laplacian of u is obtained by using \(\Delta u = \operatorname{div}\nabla u\) and noting the identities \(\nabla\rho_{l} \cdot\nabla \rho_{k} = \delta_{lk}\), \(\nabla\rho_{l} \cdot x = \rho_{l}\) and \(\Delta\rho_{l}=1/\rho_{l}\) except of course for n odd and \(l=N\), where we have \(\Delta\rho_{N}=0\). □
In the specific case \(f_{\alpha}(\rho) = {\mathscr {G}}( \|\rho\| )\) for \(\alpha= 1,\dots,k\) and with \({\mathscr {G}}\) sufficiently regular, the above quantities simplify and can be expressed as in the following lemma. Hereafter, with J as in (2.7), we write H for the constant \(n \times n\) skew-symmetric matrix
$$ \mathbf{H}= \textstyle\begin{cases} \operatorname{diag}({\mathbf{J}}, \dots, {\mathbf{J}}) &\text{if } n = 2k, \\ \operatorname{diag}({\mathbf{J}}, \dots, {\mathbf{J}}, 0) &\text{if } n = 2k+1. \end{cases} $$
(2.12)
Lemma 2.2
Let
\(u = {\mathbf{Q}}(\rho_{1}, \dots, \rho_{N}) x|x|^{-1}\)
with
\(x \in \overline{\mathbb{X}}^{n}\)
and
\((\rho_{1}, \ldots, \rho_{N}) \in\overline{{\mathbb {A}}}_{N}\)
be a spherical whirl where
Q
is as given by (2.3). Assume furthermore that
$$ f_{\alpha}(\rho) = {\mathscr {G}}\bigl(\|\rho\| \bigr), \quad\rho=(\rho _{1}, \dots,\rho_{N}) \in\overline{\mathbb{A}}_{N}, 1 \le\alpha \le k, $$
(2.13)
where we have set
\(\|\rho\|=\sqrt{\rho^{2}_{1} + \cdots + \rho^{2}_{N} } =\sqrt{x_{1}^{2} + \cdots+ x_{n}^{2}}=r\)
with
\(a \le r \le b\)
and
\({\mathscr {G}}\in {\mathscr {C}}([a,b] ,{\mathbb {R}}) \cap {\mathscr {C}}^{1}((a,b) ,{\mathbb {R}})\). Then
-
\(\displaystyle\nabla u = {\mathbf{Q}}\frac{{\mathbf{I}}_{n} + ( r\dot{{\mathscr {G}}}\mathbf{H}- {\mathbf{I}}_{n} ) \theta\otimes\theta}{r}\vspace*{3pt} \),
-
\(\displaystyle|\nabla u|^{2} = \frac{n-1}{r^{2}} + \dot{{\mathscr {G}}}^{2} |\mathbf {H} \theta|^{2} \).
Additionally, if
\({\mathscr {G}}\in {\mathscr {C}}^{2}((a,b) ,{\mathbb {R}})\), then we also have
Proof
This follows easily from Lemma 2.1 upon substituting from (2.3), (2.13) and direct differentiation. Note that in the second identity when \(n \ge2\) is even, we have \(|\mathbf{H}\theta|^{2} = 1\), whilst for n odd, we have \(|\mathbf{H}\theta|^{2} = 1 -\theta_{n}^{2}\). □
Using the description of \(|\nabla u|^{2}\) in Lemma 2.1, we can proceed by writing the \(\mathbb{F}\)-energy of a spherical whirl u as
$$\begin{aligned} \mathbb{F}\bigl[u;\mathbb{X}^{n}\bigr] &= \int_{\mathbb{X}^{n}} F\bigl(r, |\nabla u|^{2}\bigr) \, dx = \int _{\mathbb{X}^{n}}F\bigl(r, \big|\nabla\bigl[{\mathbf{Q}}(\rho_{1}, \dots, \rho_{N}) x |x|^{-1}\bigr]\big|^{2}\bigr)\,dx \\ &= \int_{\mathbb{X}^{n}}F \Biggl(r, \frac{n-1}{r^{2}} + \sum _{l=1}^{N} |{\mathbf{Q}}_{,l} \theta|^{2} \Biggr) \,dx \\ &= \int_{\mathbb{X}^{n}} F \Biggl(r, \frac{n-1}{r^{2}} + \sum _{l=1}^{k} \frac {\rho_{l}^{2}}{r^{2}} |\nabla f_{l}|^{2} \Biggr) \,dx \end{aligned}$$
(2.14)
$$\begin{aligned} &= (2 \pi)^{k} \int_{{\mathbb {A}}_{N}} F \Biggl(r,\frac{n-1}{r^{2}} + \sum _{l=1}^{k}\frac{\rho_{l}^{2}}{r^{2}}|\nabla f_{l}|^{2} \Biggr) \prod_{j=1}^{k} \rho_{j} \,d\rho =: (2 \pi)^{k} \mathbb{H}[f;{\mathbb {A}}_{N}], \end{aligned}$$
(2.15)
where the penultimate equality is obtained after a basic change of variables, and for the energy integral \(\mathbb{H}[f;{\mathbb {A}}_{N}]\) in (2.15), we have \(f = (f_{1}, \dots, f_{k})\). Indeed, the admissible vector field \(f=f(\rho)\) with \(\rho=(\rho_{1}, \ldots, \rho_{N})\) here is assumed to lie in the space
$$\begin{aligned}& \begin{aligned}[b] {{\mathscr {B}}}^{p}_{\mathsf {m}}({\mathbb {A}}_{N}) :={}& \bigl\{ f=(f_{1}, \dots, f_{k}) \in \mathscr{W}^{1,p}\bigl( {\mathbb {A}}_{N},{\mathbb {R}}^{k}\bigr): f_{l} \equiv0 \text{ on $( \partial {\mathbb {A}}_{N})_{a}$}, \\ & f_{l} \equiv2 \pi m_{l}+z_{l} \text{ on $(\partial {\mathbb {A}}_{N})_{b}$} \text{ for all $1 \le l \le k$} \bigr\} ,\end{aligned} \\& \mathsf {m}=(m_{1}, \dots, m_{k}) \in {\mathbb {Z}}^{k}, \qquad \mathsf {z}=(z_{1}, \dots, z_{k}) \in \mathbb{T}^{k}. \end{aligned}$$
(2.16)
The Euler-Lagrange equation associated with the energy integral \(\mathbb{H}[f;{\mathbb {A}}_{N}]\) from (2.15) over the space \({\mathscr {B}}_{\mathsf {m}}^{p}({\mathbb {A}}_{N})\) is seen to be the nonlinear system
$$ \textstyle\begin{cases} \operatorname{div}\{ F' (r, \frac{n-1}{r^{2}} + \sum_{l=1}^{k} \frac {\rho_{l}^{2}}{r^{2}} |\nabla f_{l}|^{2} ) \frac{\rho_{\alpha}^{2}}{r^{2}}\nabla f_{\alpha}\prod_{j=1}^{k} \rho_{j} \}=0 &\text{in } {\mathbb {A}}_{N}, \\ F' (r, \frac{n-1}{r^{2}} + \sum_{l=1}^{k} \frac{\rho_{l}^{2}}{r^{2}} |\nabla f_{l}|^{2} ) \frac{\rho_{\alpha}^{2}}{r^{2}} \partial_{\nu}f_{\alpha}\prod_{j=1}^{k} \rho_{j} = 0 &\text{on } \Gamma_{N}, \\ f=(f_{1}, \dots, f_{k}) \equiv0 &\text{on } (\partial {\mathbb {A}}_{N})_{a}, \\ f = (f_{1}, \dots, f_{k}) \equiv2\pi \mathsf {m} + \mathsf {z} &\text{on } (\partial {\mathbb {A}}_{N})_{b}, \end{cases} $$
(2.17)
where \(\alpha= 1, \dots, k\), \(\mathsf {m}=(m_{1}, \dots, m_{k})\) and \(\mathsf {z}=(z_{1}, \dots, z_{k})\). Note that \(\partial_{\nu}\) is the partial derivative in the outward pointing normal direction to \(\Gamma_{N}\).
Proposition 2.1
For each
\(\mathsf {m} = (m_{1}, \dots, m_{k}) \in {\mathbb {Z}}^{k}\)
and
\(\mathsf {z}=(z_{1}, \dots, z_{k}) \in\mathbb{T}^{k}\), the solution
\(f = f(\rho; \mathsf {m}) \in {\mathscr {C}}^{1}(\overline{{\mathbb {A}}}_{N},{\mathbb {R}}^{k}) \cap {\mathscr {C}}^{2}({\mathbb {A}}_{N},{\mathbb {R}}^{k})\)
to system (2.17) is unique. This solution is also the unique minimiser of
\(\mathbb{H}\)
with respect to its own boundary condition.
Proof
This is a result of a standard convexity argument. Indeed, in view of the growth assumption on \(F'\), minimisers of \(\mathbb{H}\) are solutions to the Euler-Lagrange system (2.17) and conversely by the uniform convexity of the integrand solutions to (2.17) are minimisers of \(\mathbb{H}\) with respect to their own boundary conditions. As a matter of fact, let f as described be a solution to (2.17) and pick \(g \in {\mathscr {B}}_{\mathsf {m}}^{p}({\mathbb {A}}_{N})\). Put \(\psi= g-f\) where \(\psi\equiv0\) on \((\partial {\mathbb {A}}_{N})_{a} \cup(\partial {\mathbb {A}}_{N})_{b}\). Then a standard convexity argument followed by an application of the divergence theorem gives
$$\begin{aligned} \Delta \mathbb{H}={}& \mathbb{H}[g; {\mathbb {A}}_{N}] - \mathbb{H}[f; {\mathbb {A}}_{N}] \\ \ge{}& \int_{{\mathbb {A}}_{N}} F' \Biggl(r, \frac{n-1}{r^{2}} + \sum _{l=1}^{k} \frac{\rho_{l}^{2}}{r^{2}} |\nabla f_{l}|^{2} \Biggr) \sum_{\alpha= 1}^{k} \frac{\rho_{\alpha}^{2}}{r^{2}} \bigl( |\nabla g_{\alpha}|^{2} - |\nabla f_{\alpha}|^{2} \bigr) \prod_{j=1}^{k} \rho_{j} \, d\rho \\ \ge{}& {-}2 \sum_{\alpha=1}^{k} \int_{{\mathbb {A}}_{N}}\operatorname{div}\Biggl[ F' \Biggl(r, \frac{n-1}{r^{2}} + \sum_{l=1}^{k} \frac{\rho_{l}^{2}}{r^{2}} |\nabla f_{l}|^{2} \Biggr) \frac{\rho_{\alpha}^{2}}{r^{2}} \nabla f_{\alpha}\prod_{j=1}^{k} \rho_{j} \Biggr] \psi_{\alpha}\,d\rho \\ &+2 \sum_{\alpha=1}^{k} \int_{\Gamma_{N}} \Biggl[F' \Biggl(r, \frac {n-1}{r^{2}} + \sum_{l=1}^{k} \frac{\rho_{l}^{2}}{r^{2}} |\nabla f_{l}|^{2} \Biggr) \frac{\rho_{\alpha}^{2}}{r^{2}} \partial_{\nu}f_{\alpha}\prod_{j=1}^{k} \rho_{j} \Biggr] \psi_{\alpha}\, d\rho \\ &+ \int_{{\mathbb {A}}_{N}} F' \Biggl(r, \frac{n-1}{r^{2}} + \sum _{l=1}^{k} \frac {\rho_{l}^{2}}{r^{2}} |\nabla f_{l}|^{2} \Biggr) \sum_{\alpha= 1}^{k} \frac{\rho_{\alpha}^{2}}{r^{2}} |\nabla\psi_{\alpha}|^{2} \prod _{j=1}^{k} \rho_{j} \,d\rho \ge0,\end{aligned} $$
where in deducing the last inequality we have noted that the first and second integrals on the left vanish due to f being a solution to (2.17). The uniqueness assertion now follows by observing that the last inequality is strict for nonzero ψ. □
As an instructive example, in case of the Dirichlet energy (with \(F(r,t) \equiv t\)), the above system decouples, and we can compute explicitly the unique solution \(f=(f_{1}, \ldots, f_{k})=f(\rho; \mathsf {m})\) to (2.17). This is then seen to be given by (\(n \ge3\))
$$ f_{\alpha}(\rho_{1}, \ldots, \rho_{N}; \mathsf {m}) =(2\pi m_{\alpha}+ z_{\alpha}) \frac{ \|\rho\|^{2-n} - a^{2-n}}{b^{2-n} - a^{2-n}}, \quad1 \le\alpha\le k. $$
(2.18)
Moreover, the spherical whirl associated with the above f is a solution to (1.6) iff in the even case \(m_{1} = \cdots= m_{k}\), \(z_{1} = \cdots= z_{k} \) and so \(f_{1} = \cdots= f_{k}\), and in the odd case \(z_{1}= \cdots=z_{k} = 0\), \(m_{1}=\cdots =m_{k} = 0\) and so \(f_{1} = \cdots= f_{k} =0\). Motivated by this observation, we now focus on the n even case with \(z_{1} =\cdots= z_{k} = z\) for \(z \in \mathbb{T}\) and \(m_{1}= \cdots= m_{k} =m\) for \(m \in {\mathbb {Z}}\). In this situation, as is stated below, the solution \(f=f(\rho_{1}, \dots, \rho_{N}; \mathsf {m})\) depends solely on \(\|\rho\|=\sqrt{x^{2}_{1} + \cdots+ x^{2}_{2N}}\).
Proposition 2.2
For
\(n \ge2\)
even and
\(m\in {\mathbb {Z}}\), system (2.17) admits a unique solution
\(f=f(\rho; \mathsf {m})\)
in
\(\mathscr{C}^{2}(\overline{{\mathbb {A}}}_{N}, {\mathbb {R}}^{k})\)
where
\(\mathsf {m}=(m, \dots, m)\)
and
\(\mathsf {z}=(z, \dots, z)\)
with
\(z \in\mathbb{T}\). Moreover, this solution
\(f=(f_{1}, \dots, f_{k})=f(\rho; \mathsf {m})\)
has components given explicitly by
$$ f_{\alpha}(\rho_{1}, \ldots, \rho_{N}; \mathsf {m}) ={\mathscr {G}}\bigl(\|\rho \|;m \bigr), \quad1 \le\alpha\le k, $$
(2.19)
where the function
\({\mathscr {G}}={\mathscr {G}}(r; m) \in\mathscr{C}^{2}([a,b], {\mathbb {R}})\)
is the solution to the boundary value problem
$$ \textstyle\begin{cases} \frac{d}{dr} [F' (r,\frac{n-1}{r^{2}} + \dot{{\mathscr {G}}}^{2} )r^{n-1}\dot{{\mathscr {G}}} ] = 0,\\ {\mathscr {G}}(a)=0, \\ {\mathscr {G}}(b) = 2\pi m + z. \end{cases} $$
(2.20)
Proof
That the boundary value problem (2.20) has a unique solution with the given degree of regularity follows by using variational methods. Indeed, thanks to the monotonicity and convexity assumptions on F, the energy integral
$$ \mathscr{G} \mapsto\mathbb{F}\bigl[\operatorname{exp}\bigl(\mathscr{G}(r) \mathbf{H}\bigr) x|x|^{-1}; \mathbb{X}^{n}\bigr]= n \omega_{n} \int_{a}^{b} F \biggl(r, \frac {n-1}{r^{2}} + \dot{ \mathscr{G}}^{2} \biggr) r^{n-1} \, dr $$
(2.21)
on \(\mathscr{B}^{p}_{m} = \{\mathscr{G} \in W^{1,p}(a, b) : \mathscr {G}(a)=0, \mathscr{G}(b)=2\pi m + z \}\) is sequentially weakly lower semicontinuous and coercive, and so the existence of a minimiser follows from an application of the direct methods. The \(\mathscr{C}^{2}\)-regularity and uniqueness of the minimiser \(\mathscr{G}\) then follows from standard convexity arguments and Hilbert’s differentiability theorem (cf., e.g., [1], pp. 57-61). Note also that from (2.20) it follows upon noting \(F'>0\) that the solution \(\mathscr{G}\) is monotone in r, that is, increasing when \(2\pi m + z>0\) and decreasing when \(2\pi m + z<0\). It thus remains to show that \(f=(f_{1}, \ldots, f_{k})\) as given satisfies (2.17). Indeed, f is easily seen to satisfy the boundary conditions on \((\partial {\mathbb {A}}_{N})_{a}\) and \((\partial {\mathbb {A}}_{N})_{b}\) and the flat parts of \(\partial {\mathbb {A}}_{N}\). Next, for \(1 \le\alpha\le k\) and \(1 \le i \le N\), a basic differentiation yields
$$ \frac{\partial f_{\alpha}}{\partial\rho_{i}} = \frac{ \rho_{i}}{r}\dot{{\mathscr {G}}}. $$
(2.22)
Furthermore, as \(n=2N\) and \(k=N\), we have
$$ |\nabla f_{\alpha}|^{2} = \sum_{i=1}^{k} \frac{ \rho_{i}^{2}}{r^{2}}\dot{{\mathscr {G}}}^{2} = \dot{{\mathscr {G}}}^{2} \quad\implies\quad \frac{1}{r^{2}}\sum_{l=1}^{k} \rho_{l}^{2} |\nabla f_{\alpha}|^{2} = \dot{{\mathscr {G}}}^{2}. $$
(2.23)
We can now verify that f is a solution to (2.17). To save space, we will from now on write \(\mathscr{H}(r) = (n-1)/r^{2} + \dot{{\mathscr {G}}}^{2}\). Then proceeding directly and using the ODE for \(\mathscr{G}\), we have
$$\begin{aligned} &\operatorname{div}\Biggl[ F' \Biggl(r, \frac{n-1}{r^{2}} + \frac{1}{r^{2}}\sum_{l=1}^{k} \rho_{l}^{2} |\nabla f_{l}|^{2} \Biggr) \frac{\rho_{\alpha}^{2}}{r^{2}}\nabla f_{\alpha}\prod_{j=1}^{k} \rho_{j} \Biggr] \\ &\quad= \sum_{i=1}^{k} \frac{\partial}{\partial\rho_{i}} \Biggl[ F' \biggl(r, \frac{n-1}{r^{2}} + \dot{{\mathscr {G}}}^{2} \biggr) \frac{\rho_{i}}{r^{3}}\dot{{\mathscr {G}}}\rho_{\alpha}^{2} \prod _{j=1}^{k}\rho_{j} \Biggr] \\ &\quad= \sum_{i=1}^{k} \Biggl\{ \frac{d}{dr}F'(r,\mathscr{H})\frac{\rho _{i}^{2}}{r^{4}}\dot{{\mathscr {G}}}\rho_{\alpha}^{2} \prod_{j=1}^{k} \rho_{j} + F'(r,\mathscr{H})\frac{\rho_{i}^{2}}{r^{4}}\ddot{{\mathscr {G}}}\rho_{\alpha}^{2} \prod_{j=1}^{k} \rho_{j} \\ &\qquad{}- 3F'(r,\mathscr{H})\frac{\rho_{i}^{2}}{r^{5}}\dot{{\mathscr {G}}}\rho_{\alpha}^{2} \prod_{j=1}^{k}\rho_{j} + F'(r,\mathscr{H})\frac{1}{r^{3}}\dot{{\mathscr {G}}}\rho_{\alpha}^{2} \prod_{j=1}^{k}\rho_{j} \\ &\qquad{}+ F'(r,\mathscr{H})\frac{\rho_{i}}{r^{3}}\dot{{\mathscr {G}}}2\rho_{\alpha}\delta _{i}^{\alpha}\prod_{j=1}^{k} \rho_{j} + F'(r,\mathscr{H})\frac{\rho_{i}}{r^{3}}\dot{{\mathscr {G}}}\rho_{\alpha}^{2} \prod_{j=1,j\neq i}^{k} \rho_{j} \Biggr\} \\ &\quad= \frac{\rho_{\alpha}^{2}}{r^{2}} \Biggl(\prod_{j=1}^{k} \rho_{j} \Biggr) \biggl\{ \frac{d}{dr}F'(r, \mathscr{H}) \dot{{\mathscr {G}}}+ F'(r,\mathscr{H})\ddot{{\mathscr {G}}}- 3 F'(r,\mathscr{H}) \frac{\dot{{\mathscr {G}}}}{r} \\ &\qquad{}+ k F'(r,\mathscr{H}) \frac{\dot{{\mathscr {G}}}}{r} + 2 F'(r, \mathscr{H}) \frac{\dot{{\mathscr {G}}}}{r} + k F'(r,\mathscr{H}) \frac{\dot{{\mathscr {G}}}}{r} \biggr\} \\ &\quad= \frac{\rho_{\alpha}^{2}}{r^{2}} \Biggl(\prod_{j=1}^{k} \rho_{j} \Biggr) \biggl\{ \frac{d}{dr}F'(r, \mathscr{H}) \dot{{\mathscr {G}}}+ F'(r,\mathscr{H})\ddot{{\mathscr {G}}}+ \frac{n-1}{r} F'(r, \mathscr{H})\dot{{\mathscr {G}}}\biggr\} = 0. \end{aligned}$$
(2.24)
The uniqueness of the solution f and the remaining minimality assertions follow from the previous proposition. □
From the description of the solution \(f=f(\rho_{1}, \dots, \rho_{N}; \mathsf {m})\) it follows that f is solely a function of the radial variable \(r=\|\rho\|\). Hence, with a slight abuse of notation, the associated spherical whirl has the form \(u= {\mathbf{Q}}(r) x|x|^{-1}\) where \({\mathbf{Q}}\in\mathscr{C}^{2}([a,b],\mathbf{SO}(n))\); indeed, \({\mathbf{Q}}(r) = \exp(\mathscr{G}(r) \mathbf{H})\) where \(\mathscr{G}=\mathscr{G}(r)\) is as in Proposition 2.2 and H is the constant \(n \times n\) skew-symmetric matrix from (2.12). It therefore follows from similar results in [2] (see also [3]) that the spherical whirl \(u = {\mathbf{Q}}(\rho _{1}, \dots, \rho_{N}; \mathsf {m}) x|x|^{-1}\) with Q as in (1.11) and f from Proposition 2.2 is a classical solution to the nonlinear system (1.6) when n is even. Alternatively and more directly, referring to (1.6), Proposition 2.2, the explicit form of \(u = \exp(\mathscr{G}(r) \mathbf{H})x|x|^{-1}\) and the ODE (2.20) satisfied by \(\mathscr{G}=\mathscr {G}(r)\), we can write, with \(\mathscr{H}(r) = (n-1)/r^{2} + \dot{{\mathscr {G}}}^{2}\) as before and starting from \(\mathscr{L}[u]\):
$$\begin{aligned}[b] \mathscr{L}[u] ={}& \operatorname{div}\bigl[ F'\bigl(r, |\nabla u|^{2}\bigr)\nabla u \bigr] + F'\bigl(r, |\nabla u|^{2}\bigr)|\nabla u|^{2} u \\ ={}& 2 F''(r, \mathscr{H}) \bigl(\dot{{\mathscr {G}}}\ddot{{\mathscr {G}}} - (n-1)r^{-3} \bigr){\dot{\mathbf{Q}}}\theta+ \partial_{r} F'(r, \mathscr{H}){\dot{\mathbf{Q}}}\theta+ F'(r, \mathscr{H}) \\ & \times \bigl({\ddot{\mathbf{Q}}}+ (n-1) r^{-2} (r{\dot{\mathbf{Q}}}- {\mathbf{Q}}) \bigr)\theta+ F'(r, \mathscr{H}) \bigl( \dot{{\mathscr {G}}}^{2} + (n-1)r^{-2} \bigr){\mathbf{Q}}\theta \\ ={}& \bigl\{ 2 F''(r, \mathscr{H}) \bigl(\dot{{\mathscr {G}}}\ddot{ {\mathscr {G}}} - (n-1)r^{-3} \bigr)\dot{{\mathscr {G}}} + \partial_{r}F'(r, \mathscr{H})\dot{{\mathscr {G}}} \\ &+ F'(r, \mathscr{H}) \bigl( \ddot{{\mathscr {G}}} + (n-1)r^{-1} \dot{{\mathscr {G}}} \bigr) \bigr\} \mathbf{H}{\mathbf{Q}}\theta=0. \end{aligned} $$
(2.25)
This proves that u is a solution to the nonlinear system (1.6) and hence justifies the Main Theorem stated in the Introduction. As a remark, this also shows that for spherical whirls u with f as in Proposition 2.2 (cf. (2.19)), the reduced system (2.17) is equivalent to the original full system (1.6), a conclusion that is not in general true for spherical whirls with unequal components of f merely solving (2.17). As an example, see (2.18) and the accompanying discussion.