on carleman estimates for elliptic and parabolic operators ...

18 JÉRÔME LE ROUSSEAU AND GILLES LEBEAU
7.2. Estimate at the boundary. If we place ourselves in the neighborhood of the boundary we have the
following result.
Theorem 7.5 (Carleman estimate at the boundary). Let x 0 ∈ ∂Ω and K be a compact set of Ω, x 0 ∈ K, and
V an open subset of Ω that is a neighborhood of K in Ω, with K and V chosen sufficiently small. Let ϕ be a
weight function that satisfies Assumption 7.1 in V, with (7.1) replaced by (7.3), and ∂ n ϕ| ∂Ω∩V < 0, where n
is the outward pointing unit normal to Ω. Then there exist C > 0 and δ 3 > 0 such that
‖h 1 2 e ϕ/h u‖ 2 L 2 (Q) + ‖h 3/2 e ϕ/h ∇ x u‖ 2 L 2 (Q) ≤ C‖h 2 e ϕ/h Pu‖ 2 L 2 (Q),
for 0 < (T + T 2 )ε ≤ δ 3 , h = εt(T − t) and u ∈ C ∞ ([0, T] × Ω), with supp(u(t)) ⊂ K for all t ∈ [0, T], and
u| (0,T)×(∂Ω∩V) = 0.
The proof of this estimate is more technical than that of Theorem 7.3. We have placed it in Appendix
A.9. The idea of the proof is to use the Gårding inequality in the tangential directions, including the
time direction. The original proof for this estimate is available in [FI96]. However, following the approach
of [FI96] does not put forward the sufficiency of the sub-ellipticity condition (7.3).
7.3. Global estimate. We now focus our attention on global Carleman estimates. We proceed by patching
together the local estimates we have presented here, in the interior and at the boundary. The global aspect
of the estimate will impose an “observation” term over (0, T) × ω, with ω ⋐ Ω in the r.h.s. of the estimate.
To patch these local estimates together we choose a global weight function that can be used to derive
each of these local estimates by satisfying the following requirements.
Assumption 7.6. Let ω 0 ⋐ ω ⋐ Ω. The weight function ϕ satisfies
ϕ| ∂Ω = Cst, ∂ n ϕ| ∂Ω < 0, sup ϕ(x) < 0, |ϕ ′ (x)| 0, x ∈ Ω \ ω 0 ,
x∈Ω
q 2 | ε=0 = 0 ⇒ {q 2 | ε=0 , q 1 | ε=0 } > 0, x ∈ Ω \ ω 0 ,
Such conditions can be fulfilled by taking ϕ of the form
ϕ(x) = e λψ(x) − e λK , with K > ‖ψ‖ ∞ , |ψ ′ (x)| 0, x ∈ Ω \ ω 0 , and
ψ| ∂Ω = 0, ∂ n ψ| ∂Ω < 0, ψ(x) > 0, x ∈ Ω,
and by taking the positive parameter λ sufficiently large. For the construction of such a function ψ we refer
to [FI96, Lemma 1.1]. The construction makes use of Morse functions and the associated approximation
theorem [AE84].
Theorem 7.7 (Global Carleman estimate). Let ϕ be a function that satisfies Assumption 7.6. Then there
exist δ 4 > 0 and C ≥ 0 such that
(
)
‖h 1 2 e ϕ/h u‖ 2 L 2 (Q) + ‖h 3/2 e ϕ/h ∇ x u‖ 2 L 2 (Q) ≤ C ‖h 2 e ϕ/h Pu‖ 2 L 2 (Q) + ‖h 1 2 e ϕ/h u‖ 2 L 2 ((0,T)×ω) ,
for 0 < (T + T 2 )ε ≤ δ 4 , h = εt(T − t) and u ∈ C ∞ ([0, T] × Ω) such that u| (0,T)×∂Ω = 0.
Proof. Let ω 1 be such that ω 0 ⋐ ω 1 ⋐ ω. For all x ∈ Ω \ ω 1 , there exist an open subset V x of Ω, with
x ∈ V x ⊂ Ω \ ω 0 for which the local Carleman estimate, in the interior or at the boundary, holds with the
weight function ϕ, for smooth functions with support in the compact K x = V x .
From the covering of Ω \ ω 1 by the open sets V x , x ∈ Ω \ ω 1 we can extract a finite covering (V i ) i∈I , such
that for all i ∈ I the Carleman estimate in V i holds for h < h i , C = C i > 0 and supp(u) ⊂ K i = V i .
Let (χ i ) i∈I be a partition of unity subordinated to the covering V i , i ∈ I, [Trè67, Hör90], i.e.,
χ i ∈ C ∞ ∑
(Ω), supp( χ i ) ⊂ K i = V i , 0 ≤ χ i ≤ 1, i ∈ I, and χ i = 1 in Ω \ ω 1
Note that we have supp( χ i ) ∩ ω 0 = ∅. For all i ∈ I, we set u i = χ i u. Then for each u i we have a local
Carleman estimate. We now observe that we have
Pu i = P( χ i u) = χ i Pu + [P, χ i ]u = χ i Pu − [∆, χ i ]u,
i∈I