Davit Harutyunyan, davith@math.utah.edu
Co-authors: Graeme W.Milton
Examples of extremal quasiconvex quadratic forms that are not polyconvex
We prove that if the associated fourth order tensor of a quadratic form has a linear elastic cubic symmetry then
it is a quasiconvex form if and only if it is polyconvex, i.e., a sum of convex and null–Lagrangian quadratic
forms. We prove that allowing for slightly less symmetry, namely only cyclic and axis–reflection symmetry,
gives rise to a class of extremal quasiconvex quadratic forms, that also turn out to be non–polyconvex.
Examples of extremal quasiconvex quadratic forms
that are not polyconvex
Davit Harutyunyan
joint with G.W.Milton (University of Utah)
University of Utah
07.24.2014
Convexity, Quasiconvexity, Rank-One Convexity
CV A function f : D → R is convex, if
f (λx + (1 − λ)y ) ≤ λf (x) + (1 − λ)f (y )
for allx, y ∈ D.
QCV A function f : R N×n → R is quasiconvex, if
Z
|D| · f (ξ) ≤
f (ξ + ∇ϕ(x)) dx,
D
for all D ⊂ Rn and ϕ ∈ W01,∞ (D, RN ).
ROCV A function f : R N×n → R is rank-one convex, if
f (λx + (1 − λ)y ) ≤ λf (x) + (1 − λ)f (y )
for all x, y ∈ R N×n with rank(x − y ) ≤ 1.
Convexity =⇒ Quasiconvexity =⇒ Rank-One Convexity
Morrey (1952).
Why quasiconvexity?
Euler-Lagrange Equation
Consider
Z
inf
u∈A
L(∇u, u, x) dx,
Ω
where Ω ⊂ Rn is an open bounded set, u : Ω → RN ,
L = L(pik , z1 , . . . , zN , x1 , . . . xn ) : RN×n × R N × R n → R, (pik =
∂ui
∂xk
) and
A = {u ∈ W 1,p : u − u0 ∈ W01,p },
where p > 1. Then If u is a point of local minimum, take any v ∈ C01 (Ω)
and denote f (t) = I (u + tv ). Then f 0 (0) = 0 and one gets the E.L.
equation:
n
X
−
(Lpk (∇u, u, x))xi +Lzk (∇u, u, x)) = 0 in Ω,
i
i=1
(k = 1, 2, . . . , N).
Rank-One Convexity, Null-Lagrangians
ROCV If L = L(∇u), then f ”(0) ≥ 0. For a specific choice of v one gets
N
n
X
X
∂ 2 L(ξ)
i,j=1 α,β=1
∂ξαi ξβj
λi λj µα µβ ≥ 0
Legendre-Hadamard condition, which is ROCV if L is C 2 .
N-L There are some L for which any smooth u is a solution to E.L.E.
Such L are called Null-Lagrangians. Then, I (u) depends only on B.C.
Determinants are Null-Lagrangians.
Theorem
If N = n, than L(p) = det(p) is a Null-Lagrangian.
Quasiconvexity
Z
inf
u∈A
L(∇u, u, x) dx,
Ω
where Ω ⊂ Rn L : RN×n × R N × Ω → R, p > 1
A = {u ∈ W 1,p : u − u0 ∈ W01,p }.
Direct Method. Choose a minimization sequence {uk } s.t.
I (uk ) → inf u∈A I (u).
Then if L satisfies appropriate growth condition, then {uk } is bdd. in
W 1,p thus
uk * u
in
W 1,p ,
for a subsequence.
Question: Whether uk * u implies I (u) ≤ lim inf I (uk )?
I (u) is called weakly lower semicontinuous in W 1,p (Ω; RN ).
Quasiconvexity
Theorem
Lat 1 ≤ p < ∞ and Ω ⊂ Rn be a bounded open set with a Lipschitz
boundary. Let L = L(ξ, u, x) : RN×n × R N × Ω → R be continuous
satisfying
0 ≤ L(ξ, u, x) ≤ g (u, x)(1 + |ξ|p ),
where g : Ω × RN → R is a non-negative continuous function. Then
Z
I (u) =
L(∇u(x), u(x), x) dx
Ω
is weakly lower semicontinuous in W 1,p (Ω; RN ) iff ξ → L(x, u, ξ) is
quasiconvex for all (x, u) ∈ Ω × RN .
Polyconvexity
PCV A function f : R N×n → R is polyconvex, if there exists a convex
function g such that
f (ξ) = g (ξ, M1 (ξ), . . . , Mr (ξ)),
where Mi (ξ) are the minors of ξ = (ξij ), 1 ≤ i ≤ N, 1 ≤ j ≤ n,
Ball(1977). Ball proved that then polyconvexity is an intermediate
condition between convexity and quasiconvexity, i.e.,
Convexity =⇒ Polyconvexity =⇒ Quasiconvexity =⇒ Rank-One
Convexity
Quadratic forms
In the case when the function f : RN×n → R is quadratic, i.e.,
f (ξ) = (Mξ; ξ),
for some symmetric matrix M ∈ R (N×n)×(N×n) , rank-one convexity of f
is actually equivalent to its quasiconvexity, Van Hove(1947). The
function f is rank-one convex if
X
f (x ⊗ y ) =
Mijkl xi yj xk yl ≥ 0,
for all
x ∈ RN , y ∈ Rn .
Quadratic forms
I
When n = 1 or N = 1 all four are the same as rank-one convexity is
the convexity for general functions f .
I
When n = 2 or N = 2, then QCV implies PCV, Terpstra(1938).
I
When n, N ≥ 3 then there are forms f (ξ) = (Mξ; ξ) that are
quasiconvex but not polyconvex, Terpstra(1938), proof is an
existence proof, not constructive.
Example:
2
2
f (ξ) = (ξ11 −ξ32 −ξ23 )2 +(ξ12 −ξ31 −ξ13 )2 +(ξ21 −ξ31 −ξ13 )2 +ξ22
+ξ33
−|ξ|2 ,
where > 0 is a small number, Serre(1981).
Extremal quadratic forms
Definition (1)
A quadratic quasiconvex form is called an extremal if one can not
subtract a rank-one form from it while preserving the quasiconvexity of
the form
Definition (2)
A quadratic quasiconvex form is called an extremal if one can not
subtract a quasiconvex form from it other than a multiple of itself modulo
Null-Lagrangians, while preserving the quasiconvexity of the form.
P
In the case n = 2 or N = 2 a rank-one form f (ξ) = ( i,j cij ξij )2 is
extremal in the sense of Def 2 but not in the sense of Def 1. So the two
definitions are not equivalent, but it is not known if Def 1 implies Def 2.
Definition (3)
A quadratic quasiconvex form is called an extremal if it is an extremal in
the sense of both Definition 1 and Definition 2
Milton(1990).
Algorithm of finding extremals, Milton(2013).
No extremals in the sense of Def 2 known.
Extremal quadratic forms: Applications
Why are extremals important?
I
G. Allaire and R.V. Kohn. Optimal lower bounds on the elastic
energy of a composite made from two non-well-ordered isotropic
materials (1994)
I
G. W. Milton and L.H. Nguyen. Bounds on the volume fraction of
2-phase, 2-dimensional elastic bodies and on (stress, strain) pairs in
composites (2011)
I
H. Kang and G. W. Milton. Bounds on the volume fractions of two
materials in a three dimensional body from boundary measurements
by the translation method (2013)
Extremal quadratic forms: Applications
If f is quasiconvex then for affine B.C. ϕ(x) = Ax on ∂Ω one has
Z
1
f (A) ≤
f (∇ϕ) dx,
|Ω| Ω
I
G. W. Milton. Sharp inequalities which generalize the divergence
theorem, (2013).
hf (E )i ≥ f (hE i)
for periodic fields E = LU, where L is a differential operator and
U = U0 + U1 , where U0 is a polynomial and U1 is a periodic field.
This gives more B.C.
Results
Theorem
Assume that f (ξ) = ξT ξ T is a quadratic form, where T is a linear elastic
cubic symmetric rank-four tensor and ξ = {ξij }3i,j=1 . Then if f is
quasiconvex it can be written as a sum of convex and Null-Lagrangian
forms (QCV =⇒ PCV)
We will write f (x, y ) = f (x ⊗ y ) for x, y ∈ R3 .
We say that f has
I
Swap symmetry if f (x, y ) = f (y , x)
I
Cyclic symmetry if f (x1 , x2 , x3 , y1 , y2 , y3 ) = f (x2 , x3 , x1 , y2 , y3 , y1 )
I
Axis-reflection symmetry if
f (x1 , x2 , x3 , y1 , y2 , y3 )
I
=
f (−x1 , x2 , x3 , −y1 , y2 , y3 )
=
f (x1 , −x2 , x3 , y1 , −y2 , y3 )
=
f (x1 , x2 , −x3 , y1 , y2 , −y3 ).
Cubic symmetry, if it has all previous three.
(1)
Results
If we drop the swap symmetry, then there is an extremal that is not
PCV!
Theorem
The quadratic form
2
2
2
2
2
2
Q(ξ) = (ξ11
+ ξ22
+ ξ33
− 2ξ11 ξ22 − 2ξ22 ξ33 − 2ξ33 ξ11 ) + ξ12
+ ξ23
+ ξ31
,
has the following properties:
(i) Q is quasiconvex
(ii) Q is not polyconvex
(iii) Q is an extremal, in all three senses of extremal.
Corollary
The quadratic form Qα,β,γ (ξ) =
2
2
2
2
2
2
(ξ11
+ ξ22
+ ξ33
− 2ξ11 ξ22 − 2ξ22 ξ33 − 2ξ33 ξ11 ) + αξ12
+ βξ23
+ γξ31
, has
the same properties as Q(ξ), where α, β, γ > 0 with αβγ = 1.
Rank-One Equivalence
Definition
Two quadratic forms f (ξ) = ξ T T ξ and g (ξ) = ξT 0 ξ T are called
rank-one equivalent if there exist nonsingular linear transformations
A, B : R3 → R3 , such that
f (x, y ) = g (Ax, By ) for all
x, y ∈ R3 .
Theorem
(i) Any quadratic form f is rank-one equivalent to itself.
(ii) If f is is rank-one equivalent to g then g is is rank-one equivalent to
f.
(iii) If f is is rank-one equivalent to g and g is is rank-one equivalent to
h then f is rank-one equivalent to h.
(iv) If the quadratic forms f and g are rank-one equivalent, then f is
quasiconvex if and only if g is so.
(v) If the quadratic forms f and g are rank-one equivalent, then f is an
extremal quasiconvex form in the sense of Definition 4 if and only if
g is so.
Ideas of Proof
I
quasiconvexity f (x, y ) = x T T (y )x, where T (y ) is a 3 × 3 matrix
the entries of which are quadratic forms in y . f (x, y ) ≥ 0 iff
T (y ) ≥ 0 for all y .
 2

y1 + y22 −y1 y2
−y1 y3
T (y ) =  −y1 y2 y22 + y32 −y2 y3  .
−y1 y3
−y2 y3 y32 + y12
Principal minors of T (y ) are nonnegative:
2
y + y32 −y2 y3
M11 = det 2
= y12 y22 + y12 y32 + y34 ≥ 0,
−y2 y3 y32 + y12
2
y + y22 −y1 y3
M22 = det 1
= y12 y22 + y22 y32 + y14 ≥ 0,
−y1 y3 y32 + y12
2
y + y22 −y1 y2
M33 = det 1
= y12 y32 + y22 y32 + y24 ≥ 0,
−y1 y2 y22 + y32
and
det(T (y )) = y14 y22 + y24 y32 + y34 y12 − 3y12 y22 y32 ≥ 0,
by the Cauchy-Schwartz.
Ideas of Proof
I
Non-polyconvexity Q is polyconvex iff
Q(η) ≥
9
X
αi Mi (η) for all
η ∈ R3×3 ,
(2)
i=1
where Mi (η) is the i-th 2 × 2 minor of η. Each minor Mi (η) contains
terms that involve one of the entries η13 , η21 or η31 while the left
hand side contains none of then, thus αi = 0 and Q(η) ≥ 0, which
contradicts Q(I ) = −3.
I
Extremality Assume Q = Q1 + Q2 with 0 ≤ Q1 (x, y ) ≤ Q(x, y ),
then Q1 = αQ.
Analysis of Q(y ), Mi (T1 (y )), det(T1 (y )) and det(T (y )).
Special Fields of Q: B.C. and Sharp Inequalities
Consider
2
2
2
2
2
2
Q(ξ) = (ξ11
+ ξ22
+ ξ33
− 2ξ11 ξ22 − 2ξ22 ξ33 − 2ξ33 ξ11 ) + ξ12
+ ξ23
+ ξ31
,
α = β = γ = 1. Then the special fields (Milton(2013)) are E = ∇u, for
which
hQ(∇u)i = Q(h∇ui).
It turns out that u must be of the form:
u1 =
v0 (x1 + x2 + x3 ) − v1 (−x1 + x2 + x3 )
+v2 (x1 − x2 + x3 ) + v3 (x1 + x2 − x3 ),
u2 =
v0 (x1 + x2 + x3 ) + v1 (−x1 + x2 + x3 )
−v2 (x1 − x2 + x3 ) + v3 (x1 + x2 − x3 ),
u3 =
v0 (x1 + x2 + x3 ) + v1 (−x1 + x2 + x3 )
+v2 (x1 − x2 + x3 ) − v3 (x1 + x2 − x3 ),
(3)
where v1 , v2 and v3 are 2-periodic C 1 functions.
Special Fields of Q: B.C. and Sharp Inequalities
Then if Ω ∈ R3 is C 1 and u = u on ∂Ω, then
Z
Z
Q(∇u(x)) dx ≥
u · Jn dS,
Ω
∂Ω
where n is the outward unit normal to ∂Ω and
 0
−v0 − v10 − v20 − v30 v00 − v10 − v20 + v30

0
−v00 − v10 − v20 − v30
J=
0
0
0
0
v0 − v1 + v2 − v3
0

0
v00 + v10 − v20 − v30 
−v00 − v10 − v20 − v30
It is sharp, being attained when u(x) = u inside Ω.
Thank You