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

436 D. Serre / Journal of Functional Analysis 236 (2006) 409–446
from which we find
v ∗ H(η)v=|η||v| 2 + 2Y · ηℑ(v3 ¯v2) + 2Z · ηℑ(v1 ¯v3) + 2V · ηℑ(v2 ¯v1). (46)
The Lopatinskiĭ determinant writes
with
−ω iV · η −iZ · η
(τ, η) =
−iV · η −ω iY · η
iZ · η −iY · η −ω
= ωq(η)− ω 2 ,
q(η) := (V · η) 2 + (Y · η) 2 + (Z · η) 2 . (47)
Let λ±(q) denote the eigenvalues of the quadratic form q. The roots of (·,η) are given by
τ =±i|η| (for ω = 0), and by
τ 2 = q(η)−|η| 2 . (48)
On the one hand, this equation has a positive real root for some η if and only if 1 1.
When λ+(q) < 1, the well-posedness of the hyperbolic IBVP is a consequence of Theorem
3.5. Alternately, we may remark that W is convexifiable in the sense of Section 2.1, and
then apply the Hille–Yosida theorem. Thanks to Theorem 2.1, and since d = 3,weonlyhaveto
verify that W(G,F3·) is positive whenever G ∈ M2×3(R) has rank one. To check this property,
we rewrite
where B is the boundary operator and
2W(∇xu) =|Bu| 2 + W1(∇yu),
W1(∇yu) =|∇yu| 2 − (V ·∇yu2 − Z ·∇yu3) 2 − (Y ·∇yu3 − V ·∇yu1) 2
− (Z ·∇yu1 − Y ·∇yu2) 2 .