on carleman estimates for elliptic and parabolic operators ...

14 JÉRÔME LE ROUSSEAU AND GILLES LEBEAU
To estimate the second term we introduce J µ = {l; µ l ≤ µ}. The Weyl asymptotics (see e.g. [Agm65]) yields
#J µ ≤ Cµ n/2 . Then, for µ > 1 large, we write
∑
e −tµ j
µ 2k ∑
j = e −tµ j
µ 2k ∑
j ≤ e −t(µ+N) (µ + N + 1) 2k ≤ ∑ #J µ+N+1 e −t(µ+N) (µ + N + 1) 2k
µ j >µ
N∈N
Nµ e −2tµ j
|φ j (y 0 )| 2 ≤ C ∑ µ j >µ e −2tµ j
µ 2k
j
≤
C ′ e √µ −C′ .
The spectral inequality of Theorem 5.4 also leads to the following unique continuation result for series
of eigenfunctions.
Proposition 5.6. Let ω ⊂ Ω be open and ε > 0. Then for all functions u = ∑ j∈N ∗ α jφ j with the complex
coefficients α j satisfying |α j | ≤ e −ε √ µ j
, j ∈ N ∗ , we have u = 0 if u| ω = 0.
This result yields an analogy between the series ∑ j∈N ∗ α jφ j and analytic functions, when the coefficients
α j satisfy the asymptotics |α j | ≤ e −ε √ µ j
.
Proof. For 0 ≤ s < ε we set v(s, x) = ∑ j∈N α sinh( √ µ j s)
∗ j √ µ j
φ j (x). The asymptotic behavior we have assumed for
the coefficients α j yields v ∈ C 2 ((0, ε), H 2 (Ω)). We then apply the interpolation inequality of Theorem 5.3
taking Y = (α, S 0 − α) × Ω with 0 < α < S 0 − α < S 0 < ε. Since v satisfies the proper boundary conditions
and since Av = 0 and ∂ s v| {0}×ω = u| ω = 0, this yields ‖v‖ H 1 (Y) = 0. For almost every s ∈ (α, S 0 − α) we thus
have x ↦→ ∑ j∈N α sinh( √ µ j s)
∗ j √ µ j
φ j (x) = 0 in L 2 (Ω). The orthogonality of the eigenfunctions gives α j = 0 for all
j ∈ N ∗ .
µ j >µ
6. CONTROL OF THE HEAT EQUATION
We shall now construct a control function for the heat equation in the time interval (0, T) for an initial
condition y 0 in L 2 (Ω),
⎧
∂ t y − ∆y = 1 ω v in Q = (0, T) × Ω,
⎪⎨
(6.1)
y = 0
on Σ = (0, T) × ∂Ω,
⎪⎩ y(0) = y 0 in Ω.
The function v is the control. The goal is to drive the solution y to zero at time T > 0, yet only acting in the
sub-domain ω.