Estimation optimale du gradient du semi-groupe de la chaleur sur le ...

The assumption tells us that the form
D. Serre / Journal of Functional Analysis 236 (2006) 409–446 415
Q(G) + 2φ(G)· Z +|Z| 2
takes non-negative values when Z ∈ R l and G has rank one. This amounts to saying that the
quadratic form
Q(G) − φ(G) 2
is non-negative over the cone of rank-one matrices of size (d − 1) × n. Since we have min{d −
1,n} 2, this implies ([16, Theorem 2.3], see also [18]) that there exists a null-form Q0 over
M(d−1)×n(R) such that Q+(G) := Q(G) −|φ(G)| 2 + Q0(G) is non-negative. Hence the form
is non-negative. ✷
W(F)+ Q0(G) = Q+(G) + φ(G)+ p(Y) 2
Example. Let us consider the energy of an isotropic elastic material, given by (5), which is
convex if and only if λ 0 and 2λ + μd 0, a condition that depends on the dimension. Rankone
convexity holds if and only if λ 0 and 2λ + μ 0. It is easy to see that in this latter case,
there exists a null-form Q0 such that W + Q0 is convex; however, this fact is useless when the
physical domain has a non-trivial boundary, as in our case. The meaningful statement is that
when λ 0 and λ + μ 0 (an intermediate assumption if d 3)), W(G,Y) is non-negative
when G has rank one, and therefore there exists a tangential null-form that can be added to W to
make it convex. For instance, in the extreme case μ =−λ (say that λ = 2), then
W(F)= 1
T
F + F
2
2 − 2(Tr F) 2
= (F12 − F21) 2 + (F23 + F32) 2 + (F13 + F31) 2 + (F33 − F11 − F22) 2
+ 4(F12F21 − F11F22). (7)
The last term of the right-hand side is a TNF, and the rest is non-negative.
2.2. Convex stored energies in general
For more general energy densities, it may happen that W is convex, even though W is not
convexifiable in the sense given above. To study the convexity of W in a systematic way, we
perform a Fourier transform v = Fyu with respect to the tangential variables. This has the effect
to decouple the tangential frequencies. The following lemma is obvious, except for the notations:
we extend W to Mn(C) as a sesquilinear form. Thus W keeps real values. Additionally, we use
the same decomposition F = (G, Y ) as above.