DIFFERENTIABILITY PROPERTIES OF ISOTROPIC ...

Published in: Duke Math. J., 104 (2000), 367–373

DIFFERENTIABILITY PROPERTIES OF ISOTROPIC FUNCTIONS # ˇ ´ MIROSLAV SILHAV Y

Abstract Let f be a function defined on the set Sym of all symmetric tensors (¨ symmetric square matrices) on a vector space of arbitrary dimension. If f is isotropic (i.e., invariant with respect to the proper orthogonal group), it has a representation É through the eigenvalues of the tensor argument A X SymØ A simple proof is given of the result by Sylvester [6] saying that f is of class C r Ù r ¨ 0Ù 1Ù Ü Ù ðÙ if and only if É is of class C r Ø Moreover, an inductive formula for the derivatives D r f is given.

1 Introduction Let Sym denote the linear space of all symmetric second-order tensors on an ndimensional real vector space Vect with scalar product. (If Vect is identified with R n Ù then Sym may be identified with the set of all symmetric n by n matrices.) A function f Ú Sym r R is said to be isotropic if f A ¨ f QAQ T  for all A X Sym and all Q proper orthogonal. An isotropic function has a representation f A ¨ É a where É is a symmetric function on R n and a ¨ a1 Ù Ü Ù an  are the eigenvalues of A with appropriate multiplicities. Clearly É a ¨ f diag a in any orthonormal basis and thus if f is of class C r Ù r ¨ 0Ù 1Ù Ü Ù ðÙ then also É is of class C r Ø Ball [1] showed that for r ¨ 0Ù 1Ù 2Ù ðÙ also the converse is true and conjectured that the converse is true for all rØ This was subsequently proved by Sylvester [6] using complex technique and detailed estimates of the derivatives of eigenvalues. Earlier, Chadwick & Ogden [2–3] gave formulas for D rf Ù r ¨ 1Ù 2Ù 3Ù in terms of É and its derivatives assuming the differentiability (see also [1]). In this note I derive the result by Sylvester by elementary means and give a recursive formula for D r f in terms of É for arbitrary rØ I also specialize these formulas derive the forms of D rf Ù r ¨ 1Ù 2Ù 3Ù equivalent to those by Chadwick & Ogden.

2 Notation Throughout, the indices iÙ jÙ k range the interval 1Ù Ü Ù n( unless stated otherwise. The direct vector notation is used [4–5]. In addition to the notation explained in Introduction, we recall that a second-order tensor A is a linear transformation from # 1991 Mathematics Subject Classification. 73B05, 73B10

3. The Main Result

2

Vect into VectÙ with the product of two tensors the composition of the linear transformations. Furthermore, Orth + denotes the proper orthogonal group and Skew the set of all skew tensors. By a basis in Vect we always mean an orthonormal basis. Let Sn be the set of all real symmetric n by n matrices. Let ei be the canonical basis in R r Ø All vector spaces are finite dimensional and real. For a vector space X we denote by F r X the vector space of all symmetric r-linear forms F Ú X ˜ Ý ˜ X r R on XØ The direct notation is used to denote the derivatives (¨ differentials) of functions f defined on a vector space X with values in RÛ thus for x X XÙ the rth derivative D r f x is a symmetric r-form on XÙ i.e., D r f x X F r X and D r f Ú X r F r XØ For each positive integer rÙ each class C r function f on X and x X R n we denote by D r f x ¨ Df xÙ Ü Ù D r f xØ A function f Ú R n r R is said to be symmetric if f Pw ¨ f w for every w X R n and every n by n permutation matrix PØ We denote by CSr R n  the set of all symmetric functions of class C r on R n and by CIr Sym the set of all isotropic functions of class C r on SymØ 2.1 Proposition The function f Ú Sym r R is isotropic if and only if there exists a symmetric function É Ú R n r R such that for each basis ei ( and each A X Sym represented by A ¨ diag aÙ a X R n Ù f A ¨ É aØ

(2.1)

The correspondence ÎÚ f w É is one-to-one between isotropic functions on Sym and symmetric functions on R n Ø This is well-known and immediate. The function É is called the representation of f Ø

3 The Main Result For each i © j we denote by W ij the skew matrix with elements Mkl where Mij ¨ −Mji ¨ 1 and Mkl ¨ 0 for all other pairs of indices. For B X Sn we denote Bij ¨ W ijB − BW ij X Sn Ø Let Ω R ⊂ R n be an open ball of radius R with center at the origin. 3.1 Lemma For each positive integer r and each É X CSr R n  there exist functions F s ċÙ É  X C r − s R n Ù F r Sn Ù s ¨ 1Ù Ü Ù rÙ such that (i) kD r − s F s ċÙ É k C 0

Ω R

² C rÙ RkD r É k C 0

Ω R

Ù

where C rÙ R is a constant independent of É and k ċ k C 0 norm on Ω RÛ (ii) if s ¨ 1Ù n

F 1 aÙ É B 1  ¨ € Éi aBii1 Ù i¨1

s ¨ 1Ù Ü Ù rÙ Ω R

(3.1)

is the supremum

a X R nÙ

B 1 X Sn Û

HijsBijs Ù

a X R nÙ

(3.2)

(iii) if 1 ° s ² rÙ n

F s aÙ É B 1 Ù Ü Ù B s  ¨ € Gis Biis + i¨1

1 2

€

1²i©j²n

B 1 Ù Ü Ù B s X Sn Ù (3.3)

3. The Main Result

3

where for each iÙ

Gis ¨ Di F s − 1 aÙ É B 1 Ù Ü Ù B s − 1 

(3.4)

and for each iÙ jÙ 1 ² i © j ² nÙ 1

1 s −1 Hijs ¨ ˆ "Di F s − 1 a t Ù É Bij Ù B 2 Ù Ü Ù B s − 1 +Ü +Di F s − 1 a t Ù É B 1 Ù Ü Ù Bij * d t 0

(3.5) with the abbreviation a t Ú¨ a + t aj − ai ei Ø Proof For a fixed r by induction on sØ For s ¨ 1 we clearly have F 1 ċÙ f  X C s − 1 Ω R and (3.1) holds with C rÙ R ¨ 1Ø Let 1 ° s ² r be given, let F s be defined by (3.3)–(3.5) and let (3.1) hold with some C rÙ R for all values of s ¨ s 0 less than our s. Since F s − 1 ċÙ f  is of class C r − s + 1 by the induction hypothesis, we see from (3.4) and (3.5) that Gis Ù Hijs are all of class C r − s Ø Moreover, a differentiation, the chain rule and the induction hypothesis (3.1) provide kD r − s Gis k C 0

Ω R

² MkD r − s + 1 F s − 1 ċÙ É B 1 Ù Ü Ù B s − 1k C 0 ² MC rÙ RkD

r

É kC0

Ω R |B

1

| Ý |B

s−1

Ω R

|Ù

for some M ³ 1 independent of É Ù i.e., kD r − s Gis k C 0

Ω R

² C 0 rÙ RkD r É k C 0

Ω R

|B 1 | Ý |B s − 1 |Ù

Ω R

² C 0 rÙ RkD r É k C 0

Ω R |B

and similarly kD r − s Hijs k C 0

1

| Ý |B s − 1 |Ù

with possibly a larger value of C 0 rÙ R ¨ MC rÙ RØ

è

3.2 Lemma Let f Ú Sym r R be an isotropic function of class C r Ø Then (i) for each AÙ B 1 Ù Ü Ù B r X Sym and Q X Orth + we have D rf AB 1 Ù Ü Ù B r  ¨ D r f QAQ T QB 1 Q T Ù Ü Ù QB r Q T Û

(3.6)

(ii) for each AÙ B 1 Ù Ü Ù B r − 1 X Sym and W X Skew D r f AWÙ AÙ B 1Ù Ü Ù B r − 1  ¨ − D r − 1 f AWÙ B 1 Ù B 2Ù Ü Ù B r − 1  −Ü − D r − 1 f AB 1 Ù B 2 Ù Ü Ù WÙ B r − 1 Ø (3.7) Here AÙ B ¨ AB − BAØ Proof (i): Differentiate f A ¨ f QAQ T  r times in the directions B 1 Ù Ü Ù B r Ø (ii): In (3.6) we replace r by r − 1 and set Q ¨ e tW Ù t X RØ A differentiation with respect to t at t ¨ 0 gives the result. è 3.3 Lemma For each r ³ 1Ù each f X CIr SymÙ each basis ei ( and each AÙ B 1Ù Ü Ù B r X SymÙ represented by the matrices A ¨ A ¨ diag aÙ a X R n Ù B 1 Ù Ü Ù B r X Sn we have D r f AB 1 Ù Ü Ù B r  ¨ F r aÙ É B 1 Ù Ü Ù B r Ø (3.8)

3. The Main Result

4

Proof By induction on rØ For r ¨ 1 (3.8) and (3.2) represent a well-known formula for the first derivative of an isotropic function (e.g., [5]). Suppose that the assertion of the lemma is true for some particular r − 1 ³ 1Ø In view of the linearity of D r f AB 1 Ù Ü Ù B r  with respect to B r it suffices to prove (3.8) only for some special choices of B r and for B 1 Ù Ü Ù B r − 1 X Sym arbitrary. Namely, it suffices to take (i) B r ª B r ¨ diag ei Ù i ¨ 1Ù Ü Ù nÙ and (ii) B r ª B r ¨ B ij Ù 1 ² i © j ² nÙ where B ij denotes the n by n symmetric matrix with elements Mkl where Mij ¨ Mji ¨ 1 and all other elements Mkl vanish. Let first B r ª B r ¨ diag ei Ø By the induction hypothesis, D r − 1 f A + λB r B 1 Ù Ü Ù B r − 1  ¨ F r − 1 a + λbÙ É B 1 Ù Ü Ù B r − 1  and a differentiation combined with the fact that F r − 1 ċÙ É  is of class C 1 by Lemma 3.1 provides n

D r f AB 1 Ù Ü Ù B r  ¨ € Di F r − 1 aÙ É B 1 Ù Ü Ù B r − 1 Biir i¨1

which is (3.8) in this special case. Let B r ª B r ¨ B ij where iÙ j is a fixed pair, 1 ² i © j ² nØ Assume first that ai © aj Ø Set W ¨ W ¨ ai − aj  − 1 W ij and note that W Ù A ¨ −B ij Ø The application of (3.7) and the induction hypothesis give D r − 1 f Aarg1 +Ü +D r − 1f Aargr − 1  D f AB Ù Ü Ù B  ¨ ai − aj r−1 F aÙ É arg1 +Ü +F r − 1 aÙ É argr − 1  ¨ ai − aj r

1

r

(3.9)

where we have identified tensors with matrices and used the notation 1 arg1 ¨ Bij Ù B 2 Ù Ü Ù B r − 1 Ù

ÜÙ

r−1 argr − 1 ¨ B 1 Ù Ü Ù Bij Ø

Let a t be as in Lemma 3.1 and note that if a ¨ Ü Ù ai Ù Ü Ù aj Ù Ü  then a t for t ¨ 1 1 equals a 1 ¨ Ü Ù aj Ù Ü Ù aj Ù Ü Ø Then for A 1 ¨ A 1 ¨ diag a 1  we have Aij ¨ 0Ù and hence from (3.7) and the induction hypothesis, 0 ¨ D r − 1 f A 1 arg1 +Ü +D r − 1 f A 1 argr − 1  ¨ F r − 1 a 1 Ù É arg1 +Ü +F r − 1 a 1 Ù É argr − 1 Ø

(3.10)

Thus the last expression in (3.9) can be rewritten as the right–hand side of (3.5) which is (3.8) in this case. Next assume that ai ¨ aj Ø The preceding part of the proof shows that D r f A ε B 1 Ù Ü Ù B r  ¨

F r − 1 a ε Ù É arg1 +Ü +F r − 1 a ε Ù É argr − 1  ε

(3.11)

for each ε © 0Ù where A ε ¨ A + ε diag ei Ù a ε ¨ a + εei Ù B 1 Ù Ü Ù B r − 1 X Sn and B r ¨ B ijØ The limit as ε r 0 of the left–hand side of (3.11) is D r f AB 1 Ù Ü Ù B r ; the limit as ε r 0 of the right–hand side exists an equals Di F r − 1 aÙ É arg1 +Ü +Di F r − 1 aÙ É argr − 1  by l’Hospital’s rule. Thus (3.8) holds also if ai ¨ aj Ø

è

4. Low-Order Derivatives

5

3.4 Remark We have proved the following alternative expression of H r Ú         

1 r−1 F r − 1 aÙ É Bij Ù B 2 Ù Ü Ù B r − 1 +Ü +F r − 1 aÙ É B 1 Ù Ü Ù Bij 

if ai © aj Ù ai − aj    r−1 1 r−1   aÙ É Bij Ù B 2 Ù Ü Ù B r − 1+Ü +Di F r − 1 aÙ É B 1 Ù Ü Ù Bij  if ai ¨ aj Ø  Di F (3.12) This formula is useful for calculating the derivatives, in contrast to (3.5), which is useful for examining the differentiability properties of Hijr Ø

Hijr ¨  

3.5 Theorem Let f Ú Sym r R be an isotropic function. Then (i) f X C r SymÙ r ¨ 0Ù 1Ù Ü Ù ðÙ if and only if É X C r R n Û (ii) for each basis ei ( and each AÙ B 1 Ù Ü Ù B r X SymÙ represented by the matrices A ¨ A ¨ diag aÙ a X R n Ù B 1 Ù Ü Ù B r X Sn we have D r f AB 1 Ù Ü Ù B r  ¨ F r aÙ É B 1 Ù Ü Ù B r Ø Proof (i): The direct implication is immediate. Let us prove the converse. Note first that (3.1) and (3.8) imply that for each r ³ 1 and each R ± 0 there exists a constant C rÙ R such that if f X CIð Sym then kD r f k C 0

Ω R

² C rÙ RkD r É k C 0

Ω R Ø

(3.13)

Let f be such that É X C r R n  and let R ± 0Ø Let  Ú Sym r R be a C ð mollifier of the form  A ¨ ψ |A|Ù A X SymÙ set ε A ¨  A/ε/ε n n + 1/2Ù A X SymÙ and for each f X CIr SymÙ r ¨ 0Ù Ü Ù let f ε A ¨ ˆ f B ε A − B d B where the integral extends over Sym and dB denotes the Lebesgue measure on SymØ Clearly, f ε is isotropic. We denote by É ε the representation of f ε and note that fε r f Ù

D r f ε r D r f Ù

Éε r É Ù

D r É ε r D r É

as

ε r 0+ (3.14)

and the convergence is uniform on compact sets. By (3.13) and (3.14) we see that for each sÙ 1 ² s ² rÙ D sf ε is a Cauchy sequence in C 0 Ω RÙ F s Sym and thus there exist M s X C 0 Ω RÙ F s Sym such that D sfε r M s

as

ε r 0+

uniformly on compact sets. Since D sf ε are the continuous derivatives of f ε Ù we have s s s ˆ f ε AD g A d A ¨ −1 ˆ D f ε Ag A d A

for each g X C0ð SymØ The limit ε r 0+ gives s s ˆ f AD g A d A ¨ ˆ M Ag A d AØ

Thus M s are the distributional derivatives of f on SymØ Since M s are continuous functions, elementary considerations show that f X C r Sym and D s f ¨ M s Ù s ¨ 1Ù Ü Ù rØ (ii): Follows from Lemma 3.3. è

6

References

4 Low-Order Derivatives Let ei ( be a basis and let AÙ B 1 Ù B 2 Ù B 3 X Sym be represented by the matrices A ¨ A ¨ diag aÙ a X R n Ù B 1 Ù B 2 Ù B 3 X Sn and let the components of a be distinct. Let f Ú Sym r R and let the subscripts attached to É denote the partial derivatives of É Ø The application of (3.8), (3.2), (3.3), (3.4), and (3.12) provides n

Df AB 1 ¨ € Éi Bii1 i¨1

if f X C 1 SymÛ further, n

D 2f AB 1 Ù B 2  ¨ € Éij Bii1 Bjj2 + iÙ j ¨ 1

Éi − Éj 1 2 Bij Bij Ù 1 ² i © j ² n ai − aj €

if f X C 2 SymÙ cf. [3, 1], and D 3 f AB 1 Ù B 2 Ù B 3  ¨

n

€

3 Éijk Bii1 Bjj2 Bkk

iÙ jÙ k ¨ 1

+

n

€

k¨1 1²i©j²n

+2

Éik − Éjk 1 2 3 2 1 Bij Bij Bkk + Bij1 Bkk Bij3 + Bkk Bij2 Bij3  ai − aj

n

€

iÙ jÙ k ¨ 1 i©j©k©i

ai Éj − Ék  + aj Ék − Éi  + ak Éi − Éj  1 2 3 Bki Bkj Bij ai − aj  aj − ak  ak − ai 

if f X C 3 SymØ The last formula is equivalent to the one given in [2].

References 1 2 3 4 5 6

Ball, J. M.: Differentiability properties of symmetric and isotropic functions Duke Math. J. 51 (1984) 699–728 Chadwick, P.; Ogden, R. W.: On the definition of elastic moduli Arch. Rational Mech. Anal. 44 (1971) 41–53 Chadwick, P.; Ogden, R. W.: A theorem of tensor calculus and its application to isotropic elasticity Arch. Rational Mech. Anal. 44 (1971) 54–68 Gurtin, M. E.: An introduction to continuum mechanics Boston, Academic Press (1981) ˇ Silhav´ y, M.: The mechanics and thermodynamics of continuous media Berlin, Springer (1997) Sylvester, J.: On the differentiability of O(n) invariant functions of symmetric matrices Duke Math. J. 52 (1985) 475–483

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