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

This polynomial satisfies
D. Serre / Journal of Functional Analysis 236 (2006) 409–446 439
Q −c 2
′ 1
S = 1, Q(0) = 0, Q (0) = 16
c2 −
S
1
c2
, Q(±∞) =+∞.
P
If λ + μ is positive, W is convexifiable by a TNF, and therefore the homogeneous hyperbolic
IBVP is well-posed. This can be checked directly by showing that does not vanish, except at
boundary points of elliptic type τ =±icR|η|, where −c2 R is the unique root of Q in the interval
(−c2 S , 0). Notice that Q′ (0) is positive in this case, so that such a root must exist. The number
cR, with the dimension of a velocity, is the speed of Rayleigh waves, the FESW of this problem.
If λ + μ is negative, then Q ′ (0) is negative, and Q must have a positive root c2 ,bythe
Intermediate Value theorem. For τ = c|η|, we then have
4|η| 2
ωSωP + 2|η| 2 +
τ 2 2 (τ, η) = 0,
λ
where the parenthesis is a positive real number. Therefore (c|η|,η) ≡ 0 and the hyperbolic
IBVP is strongly ill-posed.
Theorem 6.2. When n = d = 3 and the energy density is given by (5), with λ>0 and 2λ + μ>0
for uniform rank-one convexity, we have:
(1) The homogeneous hyperbolic IBVP is well-posed provided λ + μ>0, or equivalently
cP >cS.
(2) The hyperbolic IBVP is strongly ill-posed when λ + μ0, certainly give a control of the L2-norms of ∂iui (i = 1,...,3), ∂1u2 and ∂2u1, and
of ∂2u3 + ∂3u2, ∂1u3 + ∂3u1. But since Ω has a boundary, Korn’s inequality does not apply
and this does not give a control neither of ∂3u nor of ∇u3. As remarked in Section 3.5.1, the
coercivity of W comes from a corrector that is neither a null-from (it does yield a boundary
integral), nor differential (the matrix K does depend on η). Following Section 3.5, we may
take
λ|η|I2 i(ν − λ)η
K(η) =
i(λ− ν)ηT
λ|η|