A calibration method for estimating critical cavitation loads from below in 3D nonlinear elasticity
Jonathan J. Bevan Jonathan H. B. Deane
math.AP 49J40, 74B20

J.J. Bevan]Jonathan J. Bevan
[J.J. Bevan]Department of Mathematics, University of Surrey, Guildford, GU2 7XH, United Kingdom.  (Corresponding author: t: +44 (0)1483   682620.)
[Corresponding author]j.bevan@surrey.ac.uk
J.H.B. Deane]Jonathan H.B. Deane
[J.H.B. Deane]Department of Mathematics, University of Surrey, Guildford, GU2 7XH, United Kingdom.
j.deane@surrey.ac.uk


A calibration method for estimating critical cavitation loads]
A calibration method for estimating critical cavitation loads from below in 3D nonlinear elasticity

  In this paper we give an explicit sufficient condition for the affine map u_λ(x):=λ x to be the global energy minimizer of a general class of elastic stored-energy functionals I(u)=∫_Ω W(∇ u) dx in three space dimensions, where W is a polyconvex function of 3 × 3 matrices.  The function space setting is such that cavitating (i.e., discontinuous) deformations are admissible.  In the language of the calculus of variations, the condition ensures the quasiconvexity of I(·) at λ, where  is the 3 × 3 identity matrix.  Our approach relies on arguments involving null Lagrangians (in this case, affine combinations of the minors of 3 × 3 matrices), on the previous work <cit.>, and on a careful numerical treatment to make the calculation of certain constants tractable.   We also derive a new condition, which seems to depend heavily on the smallest singular value λ_1(∇ u) of a competing deformation u, that is necessary for the inequality I(u) < I(), and which, in particular, does not exclude the possibility of cavitation.  


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    December 30, 2023
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§ INTRODUCTION


In this paper we consider an established model of elastic material that is capable of describing cavitation, that is, of admitting energy minimizers that are discontinuous.    This phenomenon was first analysed in the setting of hyperelasticity by Ball in <cit.>;  since then, a large and sophisticated literature has developed, including but not limited to <cit.>,  part of which focuses on finding boundary conditions which, when obeyed by all competing deformations, ensure that cavitation does not occur.   It is to the latter body of work that we contribute by considering the case of purely bulk energy

    I(u)     =∫_ W(∇ u(x)) dx,

where u: →^3 represents a deformation of an elastic material occupying the domain  in a reference configuration, and where W is a suitable stored-energy function.   In the three dimensional setting, we give an explicit characterization of those affine boundary conditions of the form

    (x)=ł x,

where ł>0 is a parameter, such that the quasiconvexity inequality

    I(u) ≥ I()

holds among all suitable maps u agreeing with  on ∂.   It is by now well established that if ł is large enough, ł≥ł_crit, say,  then such an inequality cannot hold.   Thus we probe ł_crit by finding ł_0 such that (<ref>) holds whenever ł≤ł_0.  This question has been addressed in <cit.> and, more recently, in <cit.>.   In this paper we use a new approach, involving the addition of a suitable null Lagrangian (a method sometimes known as calibration), to deduce concrete lower bounds on λ_crit in the three dimensional case.    



The analysis centres ostensibly on functions of the singular values of 3 × 3 matrices.  Let A be a 3 × 3 matrix.  Then the singular values of A are normally written as ł_j(A), for j=1,2,3, and their squares are the eigenvalues of A^TA.   See <cit.> or <cit.> for useful introductions to singular values, as well as <cit.> for an illustration of their use in nonlinear elasticity.  Singular values arise naturally in the stored-energy functions of isotropic elastic materials, and also in lower bounds which can be derived from them.  Such was the case in <cit.>, where, for 2 < q < 3 and for convex functions Z and h,  a stored-energy function very similar to[The original functional contained an `artificial' quadratic term r|A|^2, with r large, to deal with the difficulties presented by the function P. This is no longer needed thanks to the calibration method we introduce.]

    W(A) = |A|^q + Z( A) + h( A)

was shown to obey the inequality

    I(u)-I()    ≥∫_κ |∇ u - ∇|^q + h'(ł^3)Π_j=1^3(ł_i(∇ u)-ł)+ł P(∇ u) dx.

The function P is defined by

    P(A) = ∑_1 ≤ i < j ≤ 3λ_i(A)λ_j(A) - λ∑_1 ≤ i ≤ 3λ_i(A)

and the constant κ satisfies bounds defined in (<ref>) below.  By grouping the first two integrands in (<ref>) together, it is possible to find conditions on ł such that ∫_κ |∇ u - ∇|^q + h'(ł^3)Π_j=1^3(ł_i(∇ u)-ł)  dx ≥ 0.  However, the corresponding inequality for P, namely 

    ∫_ P(∇ u)  dx ≥ 0,

which, since P(ł)=0, is equivalent to the quasiconvexity of P at ł, remains an open question.  We show in this paper that P does satisfy a condition necessary for quasiconvexity at ł (see Proposition <ref>, part (a): rank-one convexity at ł), but that the most tractable sufficient condition for (<ref>) cannot hold (see Proposition <ref>, part (b): polyconvexity at ł)).  Trying instead to find conditions under which each of 

    ∫_ (κ/2) |∇ u - ∇|^q + h'(ł^3)Π_j=1^3(ł_i(∇ u)-ł) dx    ≥ 0 
    ∫_ (κ/2) |∇ u - ∇|^q + h'(ł^3)P(∇ u)  dx    ≥ 0

holds is closer to the right approach, although for reasons connected with the curvature of P at ł, (<ref>) is still not possible!  This is what leads us to introduce the null Lagrangian 

    N(A)= A - ł A,

which has the property that ∫_ N(∇ u)  dx = 0 for any admissible u and is such that there are conditions on ł under which 

    ∫_ (κ/2) |∇ u - ∇|^q + h'(ł^3)(P(∇ u)-N(∇ u))  dx    ≥ 0

for all admissible u.   See Theorem <ref> and (<ref>) in particular.    In fact, N is the unique null Lagrangian for which this method works:  see Proposition <ref>.
More generally, we remark that P and G:=P-N possess properties that are both interesting in their own right and, at the same time, highly non-trivial to derive. (See Section <ref>.)   A useful introduction to null Lagrangians can be found in <cit.>.

The upper bound ł_0 given in the right-hand side of (<ref>) is investigated in Section <ref> using a careful mixture of analysis and numerical techniques.   The partnership between these approaches seems to be particularly fruitful when applied to G and to functions derived from it.  Accordingly, we find an explicit constant ν_1 ≈ 0.4501 such that if 

    0 ≤ł^3-q h'(ł^3)   ≤κ/2 (√(2))^q-3ν_1^2-q,

 then I(u) ≥ I().  See Section <ref>, Subsection <ref> and the appendices for details.

In Section <ref>, a careful analysis of the function 

    H(A)    := Π_j=1^3(ł_j(∇ u)-ł) + ł G(A)

yields, among other things, what we believe to be new necessary conditions for the inequality I(u) ≤ I().  A distinguished role seems to be played by the smallest singular value, ł_1(∇ u):  see Proposition <ref> in particular. 





  






 §.§ Notation
 
The inner product between two matrices A and B is given by A · B =  A^T B, and, as usual, A denotes the trace of A.   For a function f: ^3 × 3→ and any 3 × 3 matrix U, the shorthand 

    D_Uf(A)     =  ∇ f(A) · U  
    
    D^2_Uf(A)     = ∇^2f(A) [U,U]

will be used, where as usual ∇^2f(A)[U,U]=f_,_(ij)(kl)(A)U_ijU_kl with the summation convention in force.   When discussing polyconvexity, which is defined when it next features in the paper, we use the shorthand notation ^19 for the set ^3 × 3×^3 × 3× containing the list of minors R(A):=(A,  A,  A) of any 3 × 3 matrix.   The set of 3 × 3 square, orthogonal matrices is denoted by O(3), and the subset of O(3) consisting of those matrices with determinant equal to 1 will be written SO(3).  For any two vectors a and n in ^3, the notation a ⊗ n will denote the matrix of rank one whose (i,j) entry is a_i n_j.    Our notation for Sobolev spaces is standard.
 


§ CALIBRATION AND THE FUNCTION G(A)


In this section we give some properties of the function P and use them to derive the null Lagrangian N alluded to above.   To start with, two technical results are required.


 Let λ >0.  Then 

  (i) ∑_i=1^3 D_Uλ_i(λ) =  U;


  (ii) ∑_1 ≤ i < j ≤ 3 D_Uλ_i(λ)D_Uλ_j(λ) =  ( U)^2/2-|U|^2/4- (U^2)/4;


  (iii) λ∑_i=1^3 D_U^2λ_i (λ) + ∑_i=1^3 (D_Uλ_i(λ))^2 = |U|^2. 

In particular, 

    ∑_i=1^3λ D^2_Uλ_i(λ)     = |U|^2 -  (U^2)/2

and 

    ∑_i=1^3 (D_Uλ_i(λ))^2 = |U|^2 +  (U^2)/2.

 
 Parts (i) and (iii):  Let λ_i (ł + hU) = ł + D_Uł_i(ł) + h^2/2D_U^2ł_i(ł) + o(h^2) for each i and insert into the identity ∑_i=1^3ł_i(ł + h U) = |ł + h U|^2.  Part (i) follows by comparing terms of order h and part (iii) by comparing terms of order h^2. 



Part (ii):  Let A=ł+hU and note that, by definition, each ł_i(A) is a root z_i, say, of the polynomial (A^T A - z^2 ) = 0.   Now 
A^T A = ł^2 + 2 ł h U_s+  h^2 U^T U, so 

    0      = ( (ł^2 - z^2) + 2 ł h U_s + h^2 U^T U) 
        = τ^3 + τ^2  (2 ł h U_s + h^2 U^T U) + τ (2 ł h U_s + h^2 U^T U)  + (2 ł h U_s + h^2 U^T U),

where U_s:=(U+U^T)/2 is the symmetric part of U and τ:=ł^2 - z^2.  Using the development of ł_i(ł + h U) given above, but this time keeping only terms of order h, it follows that τ = - 2 h ł D_U ł_i(ł) + o(h^2).  Putting this into (<ref>) and writing  Z_i:= D_U ł_i(ł) for brevity,
shows that the Z_i are roots of the following polynomial equation:

    -8 ł^3 h^3 Z^3 + 8 ł^3 h^3  U Z^2 - 8 ł^3 h^3  U_s Z + 8 ł^3 h^3  U_s + o(h^3)     = 0.

Dividing by -8 ł^3 h^3, letting h → 0  and using the identity

    U_s = ( U)^2/2 - |U|^2/4 -  U^2/4

gives

    Z^3 - ( U) Z^2 +  (( U)^2/2 - |U|^2/4 -  U^2/4) Z -  U_s = 0.

The roots Z_1,Z_2,Z_3 must therefore satisfy 

    - ∑_i=1^3 Z_i     = -  U, 
    ∑_1 ≤ i < j ≤ 3 Z_i Z_j      = ( U)^2/2 - |U|^2/4 -  U^2/4 .

Replacing each Z_i with D_U ł_i(ł) in equation (<ref>) merely recovers (or provides an alternative derivation of) part (i) of the lemma, while equation (<ref>) delivers part (ii).   

Equation (<ref>) now follows by using the identity

    ∑_i=1^3 Z_i^2 = ( ∑_i=1^3 Z_i )^2 - 2 ∑_1 ≤ i < j ≤ 3 Z_i Z_j

together with parts (i) and (ii) above.  Finally, (<ref>) follows from (iii) above and (<ref>).  This concludes the proof.



Note that (<ref>) tells us, via the Cauchy-Schwarz inequality, that D^2_U(∑_i=1^3λ_i)(λ) vanishes if and only if U is a symmetric matrix.   Moreover, (<ref>) implies that ∑_i=1^3 (D_Uł_i(ł))^2 vanishes if and only if U is antisymmetric, and that in this case D_Uł_i(ł)=0 for each index i. 


  Let C_1 and C_2 be fixed 3 × 3 matrices, let C_3 be a real number, and let 

    N(A) = C_1 · (A- λ) + C_2 · ( A- λ^2 ) + C_3( A - ł^3)

for all A ∈^3 × 3.  Then 

    D_UP(λ)     = D_UN(λ)     ∀ U ∈^3 × 3

if and only if C_1, C_2 and C_3 are related by the equation

    (ł  - ł^2 C_3 - ł C_2)     = C_1 - ł C_2^T.

Moreover, 
    D^2_UP(λ)     = D^2_UN(λ)     ∀ U ∈^3 × 3

if and only if  C_2 and C_3 are related by the equation

    C_2     = (1-λ C_3) .

In particular, the unique quadratic null Lagrangian N satisfying both (<ref>) and (<ref>) is 

    N(A) =  A - λ A,

and it satisfies

    ∫_N(∇ u) dx =0

for all u belonging to W^1,2(,^3) such that u= on ∂ (in the sense of trace). 




Let A = λ + h U and note that 

    N(A)     = [ U · C_1 + ł U  C_2 - ł (U C_2) + ł^2 C_3  U ] h  + [C_2 · U + ł C_3  U ] h^2 + h^3  U.




Next, rewrite 

    P(A)     = 1/2( (∑_i=1^3λ_i(A))-λ)^2 - λ^2/2 - |A|^2/2

and, for sufficiently small h, write 

    λ_i(A)     = λ + h D_Uλ_i(λ) + h^2/2D^2_Uλ_i(λ) + ρ_i(λ,h,U),

where ρ_i is o(h^2) as h → 0.    A short calculation then yields

    P(A)     = λ h  U + λ h^2/2∑_i=1^3 D^2_Uλ_i(λ) + h^2 ∑_1 ≤ i < j ≤ 3D_Uλ_i(λ)D_Uλ_j(λ) + ρ
        = λ h  U + h^2/4(|U|^2 - (U^2))+h^2/4(2( U)^2 -|U|^2-(U^2)) + ρ
        = λ h  U + h^2/2(( U)^2 -  (U^2)) + ρ.
        = λ h  U + h^2  U + ρ.

Here, we have used the identity U = 1/2(( U)^2 -  (U^2)), (<ref>) and Lemma <ref>(ii).  The term ρ=ρ(λ,h,U) is o(h^2) as h → 0.     Comparing terms of order h in this expression with the expansion for N(A) given above, we see that D_UP(ł 1) = D_UN(ł 1) for all U if and only if (<ref>) holds.   To prove the equivalence of (<ref>) and (<ref>), simply compare terms of order h^2 to obtain

    (C_2 + ł C_3 ) · U = · U
  
for all U, and then pick U such that U= e_i ⊗ e_j.  It is then clear that (<ref>) is equivalent to (<ref>).  

Finally, to prove that N(A)= A - ł A is the unique, quadratic null Lagrangian satisfying (<ref>) and (<ref>) take C_3=0 in (<ref>) and (<ref>).  The former gives C_2=, and the latter C_1=-ł, which together imply (<ref>).    Equation (<ref>) is a standard result about null Lagrangians;  to see it without recourse to general theory, simply observe that, for sufficiently smooth φ, N(∇φ) can be written as a divergence.  The result then follows from the Green's theorem and an approximation argument. (The argument given in <cit.> serves as a useful template.) 
 
 

We remark that this establishes a simple pattern:   N(A) can apparently be obtained from P by noting that if A=Diag (λ_1,λ_2,λ_3) then P(A) =  A - λ A.   

As was pointed out in the introduction, and originally conjectured in <cit.>, it would be very useful if P were quasiconvex at the matrix ł.   
Our results in this direction are somewhat mixed.  We find that P satisfies a condition necessary for quasiconvexity at ł, but that it does not satisfy a tractable condition sufficient condition for quasiconvexity at ł.  To be precise, (a) P is rank-one convex at ł but (b) P is not polyconvex at that point.   These concepts are explained in more detail below.   We note, incidentally, that P is not globally rank-one convex.   The latter is relatively easy to see:  one can immediately calculate that, for any rank-one matrix A=a⊗ n, ł_1(A)=ł_2(A)=0 and ł_3(A)=|A|.  In particular, P(A) = -|A|, which is a concave function of A.    The foregoing discussion is summarised in the result below. 
 


 Let the function P be defined by (<ref>).  Then 


  (a) ł is a point of rank-one convexity of P, but

  (b) P is not polyconvex at λ.


 (a):  To show (a) we only need to verify that P(ł+ a ⊗ n) is convex as a function of the rank-one matrix a ⊗ n.  Without loss of generality, we may choose coordinates such that n=e_1. A calculation then shows that, if the component of a in the e_1 direction is a_1, the following expression holds:

    P(ł + a ⊗ n)     = ł (|ł+a_1|-ł).

This is clearly convex in a ⊗ n, which proves part (a) of the lemma.



(b) Assume for a contradiction that λ is a point of polyconvexity of P.   This means that there is some point (C_1,C_2,C_3) in ^19 such that

    P(A) ≥ P(λ) + C_1 · (A - λ) + C_2 · ( A - λ^2 ) + C_3( A - λ^3)

for all A in ^3 × 3.  Note that C_3 has to be zero because P is at most quadratic.  Next, take A to be a rank-one matrix such that C_1 · A=sign ((C_1)_ij)t for a given pair i,j, where t is a positive parameter to be chosen shortly.   Recall that, when A is a rank-one matrix, P(A)=-|A|.   If (C_1)_ij≠ 0, this gives

    -t    ≥ |(C_1)_ij|t - ł C_1 -ł^2  C_2,

which is easily contradicted by taking t to be sufficiently large.   Therefore C_1 = 0, leaving

    P(A)    ≥ P(ł) + C_2 · ( A - ł^2 ).

By considering A=ł + h U for arbitrary U in ^3 × 3 and small h, it is straightforward to show that this implies D_U P(ł) = D_U N̅(ł),  where 

    N̅(A)     = C_2 · ( A - ł^2 ).

In the course of Proposition <ref> it is shown that D_U P(ł)=ł U, so (<ref>) implies

    ł U     = ł ( C_2  U -  (U C_2))

for all U.  Rearranging this gives

    C_2 = ( C_2 - 1) ,

so that C_2=(1/2).    Putting this into (<ref>) gives

    P(A) ≥ A - 3 ł^2/2,

 which is easily contradicted by taking A to be of rank 1, applying the observation that P(A)=-|A| for such A, and letting |A| →∞.   This concludes the proof.


















Next, with N as in Proposition (<ref>), we define

    G(A) := P(A) - N(A)

for all 3 × 3 matrices A.  We know by equation (<ref>) in Proposition <ref> that P and N are tangent at λ, so clearly D_UG(λ) = 0 for all U.  Moreover, by (<ref>),  we also have that D^2_UG(ł)=0 for all U.
Thus G behaves like |A-λ|^3 in a neighbourhood of λ, and this is a key feature which enables us to find new lower bounds for ł_crit.  The technique for doing so is described in the next section.  We also record the following useful property of G, which flows directly from (<ref>):

    ∫_P(∇ u) dx = ∫_ G(∇ u) dx
 
for all u belonging to W^1,2(,^3) such that u= on ∂ (in the sense of trace). 






§ NEW LOWER BOUNDS ON Λ_CRIT


Let the stored-energy function W: ^3 × 3→ [0,+∞] be given by 

    W(A)    = |A|^q + Z( A) + h( A)
 
where Z: ^3 × 3→ [0,+∞) is convex and h: → [0,+∞] has the following properties:


  (H1) h is convex and C^1 on (0,+∞);

  (H2) lim_t → 0+h(t) = +∞ and lim inf_t → +∞h(t)/t > 0;

  (H3) h(t)=+∞ if t ≤ 0.

The exponent q satisfies 2 < q < 3.  Let 

    I(u) = ∫_ W(∇ u)   dx
 
and define the class of admissible maps as

    _ł={u ∈ W^1,q(;^3):  u =  on ∂,    I(u) < +∞}.

The following argument is straightforward and can be found in <cit.>.  We include it here both for completeness and as a means of deriving the function P defined by (<ref>).  Applying <cit.> to A ↦ |A|^q gives

    |∇ u|^q ≥ |ł|^q+q|ł|^q-2ł· (∇ u -ł)+κ |∇ u-ł|^q,

where

    2^2-q≤κ≤  q2^1-q  .
 
Therefore, by (<ref>) and by appealing to the convexity of Z and h, we obtain

    W(∇ u)    ≥    W(∇ u_λ) + q|ł|^q-2ł·(∇ u    - ł) + κ |∇ u-ł|^q
       +    2γł· (∇ u    - ł) + γ |∇ u - ł|^2
       +    D_AZ(ł) · (∇ u - ł)
       +    h'(ł^3)(∇ u-ł^3),

for any u ∈_ł.   Integrating (<ref>) and using the facts that both ∇ u and ∇ u are null Lagrangians in 
W^1,q(,^3) for q ≥ 2, we obtain

    I(u) - I()    ≥∫_κ |∇ u - λ|^q + h'(λ^3)λ_1 λ_2 λ_3  dx  
        = 
    ∫_κ |∇ u - λ|^q + h'(λ^3)λ̂_1 λ̂_2 λ̂_3  dx + λ h'(λ^3) ∫_ P(∇ u)  dx.

In deriving this, it may help to recall the identity

    λ_1 λ_2 λ_3 = λ̂_1λ̂_2 λ̂_3 + λ∑_1 ≤ i < j ≤ 3λ_i λ_j - λ^2 ∑_i=1^3λ_i + ł^3,

where the notation λ_i abbreviates λ_i(A) and, for each i,
λ̂_i:=λ_i - λ.   



Continuing from (<ref>), we split the first term into two equal parts and recall the property of G and P given in (<ref>), thereby obtaining:

    I(u) - I()  ≥∫_ (κ/2) |∇ u - λ|^q +     h'(λ^3)ł̂_1 ł̂_2 ł̂_3  dx  +  
          + ∫_ (κ/2) |∇ u - λ|^q  +  λ h'(λ^3)G(∇ u)  dx 
    ≥∫_ (κ/2) |Λ(∇ u) - Λ_0|^q +     h'(λ^3)ł̂_1 ł̂_2 ł̂_3  dx  +  
          + ∫_ (κ/2) |∇ u - λ|^q  +  λ h'(λ^3)G(∇ u)  dx.

Here, Λ(A) is the 3-vector with entries λ_i(A) and λ_0= (λ,λ,λ).  We have used the well-known inequality |A-λ| ≥ |Λ(A)-Λ_0|.

In keeping with the notation introduced in <cit.>, let

    _1(Λ)    =(κ/2)|Λ - Λ_0|^q + h'(λ^3)λ̂_1 λ̂_2 λ̂_3,

and, in contrast to the approach of <cit.>, let

    _2(A)    =(κ/2)|A-λ|^q + λ h'(λ^3)G(A).


In these terms we then have

    I(u)-I()    ≥∫__1 (Λ)  dx + ∫__2 (∇ u)  dx.

The sign of the first integral can be controlled by appealing to the following result:
(<cit.>)  The function _1(Λ) defined in <ref> is pointwise nonnegative on ^+++ provided h'(ł^3) > 0 and 

    (κ/2)/h'(λ^3)λ^3-q   ≥ (q-2)^(q-2)/2 q^-q/2.



The pointwise nonnegativity of _2, on the other hand, relies primarily on the argument given in Lemma <ref> below.  In short, the idea is that _2(∇ u) is dominated by |∇ u-λ|^q for both small and large values of |∇ u-λ| provided h'(λ^3) is itself not too large.  Thus we generate a new upper bound on λ which must be imposed along with (<ref>) in order to guarantee that I(u) ≥ I().  


 With G as defined in (<ref>) and for any positive constant c_0, let

    M_2(λ,c_0)     =sup{|G(A)|/|A-λ|^2: |A-λ| ≥ c_0 }
    M_3(λ,c_0)      = sup{|G(A)|/|A-λ|^3:   0 < |A-λ| < c_0 }.

 Then _2(A) ≥ 0 for all 3 × 3 matrices A provided 

    λ h'(λ^3)     ≤min_c_0{ (κ/2) max{c_0^q-2/M_2(λ,c_0), c_0^q-3/M_3(λ,c_0)}}.
 




By (<ref>), the quantity M_3(λ,c_0) is finite and, in view of the at most quadratic growth of G, M_3(ł,c_0) is uniformly bounded as a function of c_0.   M_2(ł,c_0) has the same properties, but this time we appeal to the fact that D_U G(ł)=0 for all U.  

 Let A ≠ł and let c=|A-λ|.  It is immediately clear that

    _2(A) ≥κ/2|A-ł|^q -ł h'(ł^3) |G(A)|,

and we express the right-hand side in two ways:

    κ/2|A-ł|^q -ł h'(ł^3) |G(A)|    = (κ/2|A-ł|^q-j -ł h'(ł^3) |G(A)|/|A-ł|^j)|A-ł|^j,

where j is either 2 or 3.   Now let w:=2ł h'(ł^3)/κ and suppose that (<ref>) holds.  Then, in particular, 

    w    ≤max{c^q-2/M_2(ł,c),  c^q-3/M_3(ł,c)}.

If the maximum in (<ref>) is given by c^q-2/M_2(ł,c) then w M_2(ł,c) ≤ c^q-2, and hence κ/2 c^q-2 - ł h'(ł^3) |G(A)|/c^q-2≥ 0.  Using (<ref>) with j=2, we see that _2(A) ≥ 0.  If the maximum in (<ref>) is given by c^q-3/M_3(ł,c) then we can argue similarly, this time using (<ref>) with j=3, to conclude that _2(A) ≥ 0.    


 Let f_1(c_0) = c_0^q-2/M_2(ł,c_0), f_2(c_0) = c_0^q-3/M_3(ł,c_0)
and define ξ(c_0)=max{f_1(c_0),f_2(c_0)}.   Then f_1 is nondecreasing, f_2 is nonincreasing, and 

    inf_c_0 > 0ξ(c_0)     = M_2(ł,c^*)^q-3 M_3 (ł,c^*)^2-q

where c^* is the unique fixed point of the function c_0 ↦M_2(ł,c_0)/M_3(ł,c_0).


 Let M_2(c_0)=M_2(ł,c_0) and M_3(c_0)=M_3(ł,c_0) for brevity.  It is clear from their definitions that M_2(c_0) and M_3(c_0) are nonincreasing and nondecreasing respectively.   From this and the fact that 2 < q < 3, it follows that f_1 is nondecreasing and f_2 is nonincreasing.   Since f_1(c_0) → +∞ as c_0 → +∞ and f_2(c_0) → +∞ as c_0 → 0+, there is a unique point c^* such that f_1(c^*) = f_2(c^*),  

    ξ(c_0)={[     f_2(x_0) if x_0 ≤ c^*;     f_1(c_0) if x_0 ≥ c^* ].

and where, moreover,  ξ(c^*) is the global minimum of ξ on ^+.   It is straightforward to see that the condition f_1(c^*)=f_2(c^*) is equivalent to the condition c^*=M_2(c^*)/M_3(c^*), and that f_1(c^*)=M_2(c^*)^q-3 M_3(c^*)^2-q.



We are now in a position to state the main theorem of this section.

 Let W be as in (<ref>) and suppose that λ is chosen so that 

    0 ≤λ^3-q h'(λ^3)    ≤min{ (κ/2) (q-2)^(2-q)/2 q^q/2, (κ/2)λ^2-qM_2(ł,c^*)^q-3 M_3 (ł,c^*)^2-q},

where c^* is the unique fixed point of the function c_0 ↦ M_2(ł,c_0)/M_3(ł,c_0).   Then I(u) ≥ I() for any map u ∈ H^1(,^3) whose boundary values agree with those of  in the sense of trace.  In particular, the largest possible value λ_0 satisfying (<ref>) is a lower bound for λ_crit.  Moreover, if (<ref>) holds with strict inequality then there is C=C() >0 such that 

    I(u) - I() ≥ C ∫_ |∇ u - ł 1|^q  + |G(∇ u)| dx.

for all admissible u.  

 Acccording to (<ref>), I(u)-I() is bounded below by the sum of ∫__1(∇ u)   dx and ∫__2(∇ u) dx.  By inequality (<ref>) and Lemma <ref>, the first of these integrals is nonnegative, while Lemmas <ref> and <ref> together imply that the second integral is nonnegative.   Either way, it follows that I(u) ≥ I(), as claimed.  It is then clear that λ_0, as defined above, is not larger than ł_crit.

Now suppose that (<ref>) holds with strict inequality.   The proof of <cit.> shows that the inequality ł^3-qh'(ł^3) < κ/2 (q-2)^2-q/2 q^q/2 implies, for some > 0, that  

    _1(A)    ≥ |Λ(A) - (ł,ł,ł)|^q

for all A in ^3 × 3.   Here, Λ(A)=(ł_1(A),ł_2(A),ł_3(A)) is the vector of singular values of the matrix A.   The rigidity argument, with minor modifications, given in <cit.> then shows that there is a constant, β(), say, such that

    ∫_  |Λ(∇ u) - (ł,ł,ł)|^q  dx    ≥β() ∫_ |∇ u - ł|^q  dx.
   Hence

    ∫__1(∇ u)  dx    ≥β() ∫_ |∇ u - ł|^q  dx.

Finally, we deal with the term involving _2(A).  Fix A in ^3 × 3 and let c_0 = |A-ł|.  According to the proof of Lemma <ref>, 

    _2(A) ≥κ/2(|A-ł|^q-j -2ł h'(ł^3)/κ|G(A)|/|A-ł|^j)|A-ł|^j

for j=2 and 3.   Reusing the notation w=2ł h'(ł^3)/κ, and applying the strict version of (<ref>), there is '>0, which is independent of c_0, such that 

    w+' ≤max{c_0^q-2/M_2(ł,c_0),  c_0^q-3/M_3(ł,c_0)}.
 
By rewriting (<ref>), we obtain 

    _2(A) ≥κ/2(c_0^q-j -(w+') M_j(ł,c_0))c_0^j + κ'/2|G(A)|

for j=2 and 3.   Thanks to (<ref>), the term in brackets is nonnegative, which leaves _2(A) ≥κ'|G(A)|/2.  Both terms in the right-hand side of inequality (<ref>) are now accounted for.


The goal of Theorem <ref> is to give the largest possible bound on ł such that I(u) ≥ I().    A careful look at the proof of Lemma <ref> shows that one could replace |G(A)| by G^-(A)=-min{G(A),0} and that the same conclusions would result, but with 

    M^-_2(ł,c_0)     = sup{G^-(A)/|A-ł|^2: |A-ł|≥ c_0}
    
     M^-_3(λ,c_0)      = sup{G^-(A)/|A-λ|^3:   0 < |A-λ| < c_0 }

in place of M_2(ł,c_0) and M_3(ł,c_0) respectively.  Since M_j^-(ł,c_0) ≤ M_j(ł,c_0) for j=2 and 3, it follows that the upper bound involving c^* in (<ref>) would not decrease.   Numerical evidence suggests that it does, in fact, increase, and thus provides a better (lower) bound for ł_crit.  See Remark <ref> and Section <ref> for further details.





 §.§ Calculations leading to a concrete upper bound


The upper bound (<ref>) given in the statement of Theorem <ref> contains two terms, one of which is explicitly given in terms of the exponent q and one which depends on the fixed point c^* of the function c_0 ↦ M_2(ł,c_0)/M_3(ł,c_0).   While it does not seem to be possible to find c^* purely analytically, one can nevertheless make progress using a mixture of analysis and a careful numerical calculation, as we now describe.

  With M_2(ł,c_0) as defined by (<ref>),    lim_c_0 →∞ M_2(ł,c_0) = √(2).

  First, note that the limit in the statement exists because M_2(c) is nonincreasing and bounded below.   Now 

    G(A)/|A-ł|^2    = K(Â) +ł(2  A - 3 ł)/|A-ł|^2K(Â)-ł/|A - ł|^2(∑_i=1^3ł_i(A) -  A )

where Â = A/|A| and 

    K(Â) = ∑_1 ≤ i < j ≤ 3ł_i(Â)ł_j(Â) - Â.

Let c_0 > 0.  Since K is bounded and the term ∑_i=1^3ł_i(A) -  A has at most linear growth in A, it is clear that there are constants α_1 and α_2, depending only on ł, such that

    |K(Â)| + α_1/c_0≤|G(A)|/|A-ł|^2   ≤  |K(Â)| + α_2/c_0

whenever |A-ł| ≥ c_0.  We now prove that (a) max{K(Â)}=√(2), where the maximum is taken over all unit matrices Â, and (b) that given any c>0 there is A(c) such that |A(c)-ł| ≥ c and K(A(c))=√(2).  The proposition then follows from this and (<ref>). 


To simplify the notation we replace Â by A in the following.   By the polar factorization theorem (see <cit.>), there is a positive semidefinite and symmetric matrix U and a matrix R ∈ O(3) such that A = R U.  The eigenvalues of U are the singular values ł_i(A), i=1,2,3, and one has

    K(A)     = ∑_1 ≤ i,j ≤ 3ł_i(A) ł_j(A) - ( R   U) 
        = ∑_1 ≤ i,j ≤ 3ł_i(A) ł_j(A) -( ( R)   R   Q^T   D   Q),

where the well-known decomposition U=Q^T D Q has been used, with D = diag (ł_1(A),ł_2(A),ł_3(A)) and Q in O(3). Note that R = ± 1.     It follows that 

    K(A)     = ∑_1 ≤ i,j ≤ 3ł_i(A) ł_j(A) -(P  diag (ł_2ł_3,ł_1ł_3, ł_1 ł_2)),

where P:=Q  R   R   Q^T belongs to SO(3).    The term involving the trace satisfies 

    (P  diag (ł_2ł_3,ł_1ł_3, ł_1 ł_2) = P_11ł_2ł_3 +P_22ł_1ł_3 + P_33ł_1 ł_2.

The term ∑_1 ≤ i,j ≤ 3ł_i(A) ł_j(A) is independent of P, so we can vary P in order to minimize the term involving the trace given above.    Since P belongs to SO(3), its rows and columns are orthogonal unit vectors.  In particular, |P_11| ≤ 1, so to make P_11ł_2 ł_3 minimal we should take P_11 = -1. (Here we have used the ordering ł_3 ≥ł_2 ≥ł_1.)   Hence row 1 of P is (-1,0,0), forcing the second and third rows to take the form 
(0,P_22,P_33) and (0,P_32,P_33) respectively.   In particular, P = -(P_22P_33-P_23P_32) = 1.  Without loss of generality, we may take P_22=cosα, P_32=P_23=sinα, P_33=-cosα for some α, in which case

    P_11ł_2ł_3 +P_22ł_1ł_3 + P_33ł_1 ł_2    ≥ -ł_2 ł_3 + cosα (ł_1 ł_3 - ł_1ł_2) 
       ≥ -ł_2 ł_3 - ł_1 (ł_3 - ł_2)
 
by choosing α = π.  Hence

    K(A) ≤ 2 ł_3 (ł_1 + ł_2)

where the matrix P yielding this upper bound is given by P=diag(-1,-1,1).   Bearing in mind that |A|^2 = ∑_i=1^3ł_i^2 = 1, it can be shown that 2 ł_3 (ł_1 + ł_2) ≤√(2), and that a maximizing choice of singular values is ł_1=ł_2=1/2, ł_3 = 1/√(2).  A suitable choice for a maximizing A would therefore be A_0 = P  diag (1/2,1/2,1/√(2)) = diag  (-1/2,-1/2,1/√(2)), and one can check that K(A_0)=√(2).  To conclude the proof, it is enough to choose A(c):=r A_0 for any r=r(c) large enough that |A(c) -ł| ≥ c.


Since D^2G(ł)=0, it can easily be shown that M_2(ł,c_0) is constant as a function of c_0 for all sufficiently small and positive c_0.   We are therefore justified in writing M_2(ł,0):=lim_c_0 → 0M_2(c_0), and in fact 

    M_2(ł,0) = sup{|G(A)|/|A-ł|^2:  A ≠ł}.

According to Proposition <ref>, we must have 

    M_2(ł,0)    ≥ M_2(ł,c_0) ≥√(2) for all c_0 > 0.
  Using a `brute force' approach, which we describe in the appendix, we find that √(2) - M_2(ł,0) ≈ 2.7× 10^-9, independently of ł in the range [1,2].   Thus, in view of (<ref>), it seems that M_2(ł,c_0)≡√(2) for all c_0 and ł, and we record this as:
For all ł >0 and c_0>0, M_2(ł,c_0)=√(2).


In order to explain the method used to approximate M_2(ł,0), we introduce the notation 

    m_l(A, λ) := |G(A)|/|A - λ1|^l

for l=2 and 3, and we restrict attention to ł in the range [1,2].   Choose a random starting matrix A_1,  compute g_1:=∇ m_2(A_1,ł) and find the scalar σ_1, say, which maximises m_2(A_1+σ g_1).  Let the maximising value of σ be σ_1;  then we set A_2 = A_1 + σ_1 g_1. Proceeding iteratively, we
compute a sequence of matrices A_i. It turns out that m_2(A_i, λ)
tends to √(2) as i increases, regardless of λ.


Supposing that Conjecture <ref> is right, we are now required to find M_2(ł,c^*)^q-3 M_3 (ł,c^*)^q-2=2^q-3/2M_3(ł,c^*), where c^* satisfies
c^* = √(2)/M_3(ł,c^*).   We begin by recalling that c_0 ↦ M_3(ł,c_0) is nondecreasing, and that, thanks to the at most quadratic growth of the function G(A), there is c_1(ł) such that M_3(ł,c_0)=M_3(ł,c_1) for all c_0 ≥ c_1.   We are therefore justified in defining M_3(ł,∞):=M_3(ł,c_1).   
Appealing again to numerical techniques, which we describe in the appendix, we find that, for ł in the range [1,2], there is very good agreement between c_1(ł) and the expression ν_3ł + ν_2, where ν_2 ≈ 1.764× 10^-3 and 
and ν_3 ≈ 1.842. Moreover, for the same range of ł, there is strong numerical evidence for the approximation M_3(ł,∞) ≈ν_1/ł, where ν_1 ≈ 0.4501. See Fig 1. 

 

This leads naturally to the following:

 M_3(ł,c_0) = M_3(ł,c_1)=ν_1/ł for all c_0≥ c_1(ł) and for ł in the range [1,2], where c_1(ł)=ν_3ł + ν_2 and the values of ν_1,ν_2 and ν_3 are given above. 

Let us now suppose that Conjectures <ref> and <ref> are correct.   Note that then the function p: c ↦ M_2(ł,c)/M_3(ł,c)=√(2)ł/ν_1 is independent of c for all c ≥ c_1.   There are thus two possibilties for the fixed point c^* of p:  either c^*=√(2)ł/ν_1 or c^* < c_1.   Suppose for a contradiction that c^*< c_1.   Then M_3(ł,c^*)≤ M_3(ł,c_1), and so p(c^*) ≥ p(c_1).  But p(c^*)=c^* and p(c_1)=√(2)ł/ν_1, which gives c^* ≥√(2)ł/ν_1.  By hypothesis, c_1>c^*, which when combined with the preceding inequality implies c_1 > √(2)ł/ν_1.  Applying Conjecture <ref> and rearranging, we see that this is equivalent to ν_1ν_2 ≥ (√(2)-ν_3ν_1)ł, which can only hold if ł≤ 1.357 × 10^-3.   But we supposed that ł≥ 1, which is a contradiction.    In summary, we have shown the following result.

 Let ł belong to the interval [1,2] and suppose that Conjectures <ref> and <ref> are correct.  Then the unique fixed point of the function p(c):=M_2(ł,c)/M_3(ł,c) is given by c^* =√(2)ł/ν_1, and 

    M_2(ł,c^*)^q-3 M_3(ł,c^*)^2-q=1/√(2)(√(2)ł/ν_1)^q-2.



Referring back to the upper bound given in (<ref>), we now have the following:

  Let the assumptions of Proposition <ref> hold.  Then I(u) ≥ I() provided

    0    ≤λ^3-q h'(λ^3)  ≤  (κ/2)(√(2))^q-3ν_1^2-q.



  It is enough to show that

    min{(q-2)^(2-q)/2 q^q/2,λ^2-qM_2(ł,c^*)^q-3 M_3 (ł,c^*)^2-q}    = (√(2))^q-3ν_1^2-q.

By Proposition <ref>, we clearly have λ^2-qM_2(ł,c^*)^q-3 M_3 (ł,c^*)^2-q =  (√(2))^q-3ν_1^2-q.  Therefore it remains to show that  (q-2)^(2-q)/2 q^q/2 > (√(2))^q-3ν_1^2-q for 2 < q <3.  Let z(q)=(√(2))^q-3ν_1^2-q and y(q)=
(q-2)^(2-q)/2 q^q/2.  Note that z is convex, while a short calculation reveals that y is concave on the interval (2,3).  Therefore the inequality z(q) < y(q) will follow from the pair of inequalities z(2)<y(2+) and z(3)<y(3), both of which are easy to check.  This concludes the proof.  


Corollary <ref> suggests that the bound is not the best possible:  we would really expect both terms in the bound given in (<ref>) to play a role.  One way to achieve this might be to use the quantities M_j^- in place of M_j for j=2,3.   Indeed, we find, again numerically, that there is very good agreement between M_3^-(ł,∞) and ν_1'/ł for ł in the range [1,2], where ν_1' = 0.1923.  Interestingly, if we then replace M_3 by M_3^- everywhere in the preceding calculations, we find that z(3) < y(3) no longer holds.   In other words, both terms in the upper bound given by (<ref>) appear to contribute.  This observation comes with some caveats, however; see Section <ref> in the appendix.





§ A CONDITION FOR THE INEQUALITY I(U) ≤ I()


The results in the previous sections provide conditions on ł under which the inequality I(u) ≥ I() holds for admissible maps.  It is natural to ask what information results from supposing that I(u)≤ I(), and, of our results, Theorem <ref> is the first place to look.  Now, if (<ref>) holds with strict inequality, then  sits in a `potential well', as expressed by the estimate (<ref>), which we recall here for the reader's convenience:

    I(u) - I() ≥ C ∫_ |∇ u - ł 1|^q  + |G(∇ u)| dx.

Thus, in these circumstances, I(u) ≤ I() is impossible.  If (<ref>) holds with equality then a similar remark applies, but with the additional possibility of losing one or both terms in the right-hand side of (<ref>).  And when (<ref>) fails, the preceding analysis tells us nothing about those u whose energy satisfies I(u) ≤ I().  Therefore a different approach is called for. 

Consider the following simplified model, in which we set the function Z appearing in (<ref>) to zero.   Thus we let

    W(A)     = |A|^q + h( A)   for A ∈^3 × 3,

where, as before, 2 < q < 3.    Let ł > 1 and note that, by the convexity of h, 

    I(u) - I()    ≥∫_ h'(ł^3)(∇ u - ł^3) + |∇ u|^q - |∇|^q  dx 
       ≥∫_ h'(ł^3)(∇ u - ł^3) + κ|∇ u-∇|^q dx,

where we have used (<ref>) and (<ref>), and where the constant κ obeys the bounds specified in inequality (<ref>).
Using identity (<ref>) and the definition (<ref>) of P, write 

    ∇ u -ł^3     = ł̂_1ł̂_2ł̂_3 + ł P(∇ u),

where ł̂_i:=ł_i-ł and ł_i:=ł_i(∇ u).   Finally, recall that the function G defined in (<ref>) satisfies

    ∫_ G(∇ u)  dx      = ∫_ P(∇ u)  dx

whenever u is admissible.   Combining this with (<ref>) and (<ref>), we have

    I(u) - I()    ≥∫_ h'(ł^3)(ł̂_1ł̂_2ł̂_3 + ł G(∇ u)) + κ|∇ u-∇|^q dx.

It will be useful to have a shorthand for the function with prefactor h'(ł^3) appearing in (<ref>);  accordingly, let

    H(A)     = ł̂_1(A)ł̂_2(A)ł̂_3(A) + ł G(A).


We now give a series of results which allow us to find a lower bound on the function H.  

 Let A be a 3 × 3 matrix such that A > 0 and whose singular values obey ł_1(A) ≤ł_2(A) ≤ł_3(A).  Then there is R in SO(3) such that 

    G(A)     =(1-R_11)α + (1-R_22)β + (1-R_33)γ,
 
where α = ł_2 ł_3-łł_1, β = ł_1 ł_3 - łł_2 and γ = ł_1 ł_2 - łł_3, and ł_j=ł_j(A) for j=1,2,3.


  For brevity, write ł_j in place of ł_j(A) for j=1,2,3.  Note that P(A) = α + β + γ by definition, and that, by polar decomposition, there are matrices Q_1 and Q_2, belonging to O(3), such that A= Q_2 Q_1^T D Q_1 and where D is a diagonal matrix with entries ł_1, ł_2 and ł_3.   (See <cit.>.)  Since A > 0, Q_2 must belong to SO(3).   We see that

    N(A)     =  (Q_2 Q_1^T D Q_1) - ł Q_2 Q_1^T D Q_1 
        =  (R ( D - ł D)),

where R:=Q_1 Q_2 Q_1^T belongs to SO(3).   Hence N(A) = R_11α + R_22β + R_33γ, and (<ref>) follows.  


Our aim is to minimize G(A) by allowing R to vary in SO(3).  To that end, consider the following.  

 Let R belong to SO(3) and suppose that it minimizes 
    g(R)   = (1-R_11)α + (1-R_22)β + (1-R_33)γ.
  Then 

    α R_12 -β R_21    = 0 
    α R_13 -γ R_31    = 0 
    β R_23 - γ R_32    = 0.

If none of α,β,γ is zero then 

    ((α/β)^2-1)R_12^2 + ((α/γ)^2-1)R_13^2     = 0 
    ((β/γ)^2-1)R_23^2 - ((α/β)^2-1)R_12^2     = 0.

Moreover, if exactly one of α, β, γ is zero, then  either R_11^2=1 or R_33^2=1.   The same is true if none of α, β and γ is zero, provided at least one of α > β and β > γ holds. 

  We first show that (<ref>)-(<ref>) hold.  It is well known that the tangent space to SO(3) at R consists of those ρ in R^3 × 3 such that R^T ρ is antisymmetric.   From this, it easily follows that there are real numbers a,b and c such that 

    ρ_11    = -b R_13 - a R_12
    ρ_22    = a R_21 - c R_23
    ρ_33    = b R_31  + c R_32.

Now suppose that R() is a smooth path of matrices belonging to SO(3), and satisfying R(0)=R and Ṙ(0)=ρ.   We then have

    ∂__ = 0 g(R())   = a(R_12α - R_21β) + b (R_13α - R_31γ) + c(R_23β -R_32γ),

so that, by varying a,b and c independently, the stationarity conditions (<ref>)-(<ref>) follow.  

To prove the last part of the statement, we consider cases as follows.   



Case (i): α=0 > β≥γ.
Using (<ref>) and (<ref>), we see that R_21=R_31=0.  Hence, since the first column of R is a unit vector, we must have R_11^2=1. 



Case (ii):  α> 0=β > γ.
Using (<ref>) and (<ref>), we must have R_12=0 and R_31=0.  By (<ref>), we then have R_13=0.  Hence R_11^2=1, as before.



Case (iii):  α≥β > γ=0.  
Equations (<ref>) and (<ref>) imply respectively that R_13=0 and R_23=0.  Hence R_33^2=1.



Case (iv):  α >  β > γ; α≠ 0, β≠ 0, γ≠ 0.   In this case, (<ref>) implies that R_12 = R_21 = 0, and so R_11^2=1.



Case (v):  α = β > γ; α≠ 0, β≠ 0, γ≠ 0.  Now (<ref>) implies that R_13=0 and (<ref>) that R_23=0.  Hence R_33^2=1.



Case (vi):  α >  β = γ; α≠ 0, β≠ 0, γ≠ 0.   Equations (<ref>) and (<ref>) imply that R_12=R_13=0, and hence R_11^2=1.


The next result will enable us to deal with the case α = β = γ. 

 Let R belong to SO(3) and suppose that R is symmetric.  Then 3 ≥ R ≥ -1.

If R is a symmetric, orthogonal matrix then R^2=, from which it follows that any eigenvalue μ_i of R must satisfy μ_i^2=1.  Moreover, μ_1 μ_2 μ_3=1, from which it follows that 3 ≥ R = μ_1+μ_2+μ_3 ≥ -1. 


  Let R belong to SO(3).

    g(R)= (1-R_11)α + (1-R_22)β + (1-R_33)γ

satisfies 

    g(R)    ≥min{2(β + γ),0}.


  If two or more of α, β and γ are zero then the lower bound g(R) ≥ 2(β + γ) is trivial.  Thus, to minimize g, we may begin by supposing that the conditions of Lemma <ref> apply, so that, in all cases except α = β = γ, we have either that R_11^2=1 or R_33^2=1.     First suppose that R_11^2 = 1.  Then the diagonal elements of R are either of the form 1,cosσ, cosσ for some σ, or else of the form -1, cosσ, - cosσ.  In the former case, 

    g(R)  = (1-cosσ)(β + γ).

If β+ γ≥ 0 then clearly g(R) ≥ 0. If β + γ <0 then to minimize g we take cosσ = -1 and the claimed lower bound follows.  If R_11=-1, then 

    g(R) = 2 α + β + γ + (γ - β) cosσ,

which, since γ≤β, implies that we should take cosσ = 1 in order to minimize g.  Hence 

    g(R)    ≥ 2 (α + γ) ≥ 2(β + γ).

If R_33^2=1 then the argument needed is similar.  Finally, let us suppose that α = β = γ.  Then 

    g(R) = (3 - R) γ.

If γ < 0 then g is minimized when R = -1, according to Lemma <ref>.  Hence, in this case, g(R) ≥ 4 γ = 2(β + γ).  Otherwise, g(R) ≥ 0 because R ≤ 3 by Lemma <ref>.     This completes the proof.




  Let ł > 0, let H be given by (<ref>) and let ł_1 ≤ł_2 ≤ł_3 be the singular values of A.   Then 

    H(A)    ≥{[ (ł_1-ł)(ł_2-ł)(ł_3-ł)             if ł_1 ≥ł; (ł_1-ł)(ł_2+ł)(ł_3+ł)            if ł_1 ≤ł. ].


 First suppose that ł_1 ≥ł.   Note that β+γ= (ł_1 - ł)(ł_2+ł_3) is then nonnegative, and so, by inequality (<ref>) in Lemma <ref>,  we have G(A)=g(R) ≥ 0.  Hence H(A)=(ł_1-ł)(ł_2-ł)(ł_3-ł) + ł G(A) ≥ (ł_1-ł)(ł_2-ł)(ł_3-ł).   

Now suppose that ł_1 ≤ł.  Then the lower bound in (<ref>) is 2(β+γ)≤ 0, and so 

    H(A)    ≥ (ł_1-ł)(ł_2-ł)(ł_3-ł) +2ł (ł_1-ł)(ł_2+ł_3) 
        = (ł_1-ł)(ł_2+ł)(ł_3+ł).




The main result of this subsection is the following.  
 Suppose h'(ł^3) ≥ 0 and let W be given by (<ref>).  Then any admissible map u ≠ is such that: 

    I(u) -I()    ≥∫_κ|∇ u-∇|^q  + h'(ł^3) H(∇ u)  dx.

In particular, if ∫_H(∇ u)  dx ≥ 0 then I(u) > I(), while if I(u) ≤ I() then 

    ∫_{x ∈:   ł_1(∇ u (x)) ≥ł}h'(ł^3)(ł_1-ł)(ł_2-ł)(ł_3 - ł) +    κ|∇ u-∇|^q  dx  
       ≤∫_{x ∈:   ł_1(∇ u (x)) ≤ł} h'(ł^3) (ł - ł_1)(ł+ł_2)(ł+ł_3) dx.


  This follows from (<ref>) and Proposition <ref>. 



We remark that the results of Section <ref> imply that inequality (<ref>) ought not to be possible for ł such that ł^3-qh'(ł^3) is sufficiently small (see (<ref>)).  It is not immediately obvious from (<ref>) why this should be so;  nor is it clear why such a prominent role is played by the smallest singular value ł_1(∇ u).  This surely warrants further investigation.



§ APPENDIX

 
Two different algorithms have been used to compute the quantity M_3(ł,∞):=lim_c →∞M_3(ł,c), as defined in Subsection <ref>.  We recall the notation 

    m_l(ł,A)=|G(A)|/|A-ł|^l

for l=2,3.  The algorithms are also brought to bear on the problem of calculating M_3^-(ł,∞), and we summarise the results below.



 §.§ Algorithm A: conjugate gradient


This is a `brute force' approach, which consists of 


  * Choosing a value of λ∈[1, 2];

  * Generating matrices A with all elements a_ij being uniformly
distributed random numbers over the interval [-α, α],
for given α;

  * Using the Polak-Ribiere variant of the Fletcher-Reeves algorithm <cit.>, one of the
so-called conjugate gradient methods for maximisation of smooth functions, starting 
from each of these matrices, in order to find a candidate matrix for A_l^*(λ), which is an approximate maximiser of m_3(·,ł).  

We choose α = 5, the justification for which is as follows.    Five thousand matrices were generated and those which gave the 15 largest values of m_3(ł,·) were saved as the computation proceeded.  Of these, none had an element whose modulus exceeded 3.1, hence reassuring us that the choice α = 5 is `safe' for ł in the range [1,2].


Since the algorithm is iterative, a stopping condition is required,
and this is that

    |m^(i+1) - m^(i)| ≤ϵ/2(|m^(i+1)| + |m^(i)|),

where m^(i) is the value of m_3(A_i, λ) at the i-th iteration
and ϵ = 10^-9.

Various data are saved as the computation progresses, including the current maximising
matrix, which is the latest approximation to A_l^*(λ). For half the
simulations, the initial random matrix is symmetric,
and for the other half it is not — we do not know, a priori, whether
A_l^*(λ) will be symmetric or not.  The numerics strongly indicated that A_3^*(ł) will indeed be symmetric, at least for ł in the range [1,2].

All computations were carried out using 40 significant figures.  Algorithm A leads the approximation M_3(ł,∞)≈ν_1^A/ł, where ν_1^A=0.4501 —see  Table <ref>; the algorithm also produces the approximation to c_1(ł) shown in Figure 2 and summarised in Table <ref> below.  








 §.§ Algorithm B: pointwise supremum


This is based on a different idea, although a Monte Carlo approach
it is still at its heart.    We start by fixing an interval for λ, Λ = [λ_-, λ_+], which is not
necessarily [1, 2] — the computation time is, at one level, independent of the interval. We
then define N_p equally-spaced points in Λ, these points being
λ_i = λ_- + iδλ with δλ = (λ_+ -λ_-)/N_p
and i = 0, …, N_p.

Next, as before, a large number, N, of random matrices A_i are generated. As can
be seen from its definition,

    m_l(A_i, λ) = |a_1 + a_2λ|/(b_1 + b_2λ + b_3λ^2)^l/2

where the coefficients a_1,… b_3 are functions of the elements of A_i that we compute
numerically. We define f_l, i(λ) := m_l(A_i, λ), and clearly, once the coefficients
have been computed, f_l, i(λ) can easily be found for any λ.
 
We then compute

    F_l, j = sup{f_l,i(λ_j), i = 1,…, N}

for j = 0,…, N_p; F_l, j is then a discrete approximation to 
M_l(λ_j,∞).   The convergence to M_3(ł,∞) is quite slow, but nonetheless,
choosing N large enough gives reasonable agreement with results
produced by Algorithm A, thereby providing an independent check.  Compare Fig. 1 with Fig. 2 below.

 




 §.§ Calculating M^-_3(ł,∞)
 

Using Algorithm A, the methodology is the same as for M_3(λ,∞), with the same number of random matrices generated, whose elements have
the same bounds. The investigations lead us to conjecture that

    M^-_3(λ,∞) ≈ν_1^A,-/λ,
 where ν_1^A,-≈ 0.1923.  See Figure <ref>. 


 

 


Recall that in the case of M_3(ł,∞) it was possible to compute and then model the quantity c_1(ł) accurately on the interval 1 ≤ł≤ 2 using an affine function of ł.  The same cannot be said of the corresponding quantity c^-_1(ł), and indeed this seems to behave somewhat erratically as a function of ł.    Thus the analysis leading up to
Proposition <ref> does not apply, and hence the caveat regarding the substitution of M^-_3(ł,∞)≈ν_1'/ł promised in Remark <ref>.  


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