on carleman estimates for elliptic and parabolic operators ...

10 JÉRÔME LE ROUSSEAU AND GILLES LEBEAU
S
f (x) = f (x 0 )
Ω ε
W
∇ f
x 0
ϕ(x) = ϕ(x 0 ) − ε
∇ϕ
B 0
V ′′ ϕ(x) = ϕ(x 0 )
V
V ′
Figure 3: Local geometry for the unique continuation problem. The striped region contains the support of
[P, χ]u.
We call W the region {x ∈ V; f (x) ≥ f (x 0 )} (region beneath { f (x) = f (x 0 )} in Figure 3). We choose V ′
and V ′′ neighborhoods of x 0 such that V ′′ ⋐ V ′ ⋐ V and we pick a function χ ∈ Cc ∞ (V ′ ) such that χ = 1 in
V ′′ . We set v = χu and then v ∈ H0 2 (V). Observe that the Carleman estimate of Theorem 3.5 applies to v by
Remark 3.6. We have
Pv = P( χu) = χ Pu + [P, χ]u,
where the commutator is a first-order differential operator. We thus obtain
(
)
h‖e ϕ/h χu‖ 2 0 + h 3 ‖e ϕ/h ∇ x ( χu)‖ 2 0 ≤ C h 4 ‖e ϕ/h χg(u)‖ 2 0 + h 4 ‖e ϕ/h [P, χ]u‖ 2 0
)
≤ C
(h ′ 4 ‖e ϕ/h χu‖ 2 0 + h 4 ‖e ϕ/h [P, χ]u‖ 2 0 , 0 < h < h 1 .
Choosing h sufficiently small, say h < h 2 , we may ignore the first term in the r.h.s. of the previous estimate.
We then write
h‖e ϕ/h u‖ 2 L 2 (V ′ ) + h 3 ‖e ϕ/h ∇ x u‖ 2 L 2 (V ′ ) ≤ h‖e ϕ/h χu‖ 2 0 + h 3 ‖e ϕ/h ∇ x ( χu)‖ 2 0 ≤ Ch 4 ‖e ϕ/h [P, χ]u‖ 2 L 2 (S ), 0 < h < h 2 ,
where S := V ′ \ (V ′′ ∪ W), since the support of [P, χ]u is confined in the region where χ varies and u does
not vanish (see the striped region in Figure 3).
For all ε ∈ R, we set V ε = {x ∈ V; ϕ(x) ≤ ϕ(x 0 ) − ε}. There exists ε > 0 such that S ⋐ V ε . We then
choose a ball B 0 with center x 0 such that B 0 ⊂ V ′′ \ V ε and obtain
e inf B 0 ϕ/h ‖u‖ H 1 (B 0 ) ≤ Ce sup S ϕ/h ‖u‖ H 1 (S ), 0 < h < h 2 .
Since inf B0 ϕ > sup S ϕ, letting h go to zero, we obtain u = 0 in B 0 . We have thus proven the following local
unique-continuation result.
Proposition 4.1. Let g be such that |g(y)| ≤ C|y|, x 0 ∈ Ω and u ∈ H 2 loc
(Ω) satisfying Pu = g(u) and u = 0
in {x; f (x) ≥ f (x 0 )}, in a neighborhood V of x 0 . The function f is defined in V and such that |∇ f | 0 in a
neighborhood of x 0 . Then u vanishes in a neighborhood of x 0 .
With a connectedness argument we then prove the following theorem.
Theorem 4.2 (A. Calderón theorem). Let g be such that |g(y)| ≤ C|y|. Let Ω be an connected open set in R n
and let ω ⋐ Ω, with ω ∅. If u ∈ H 2 (Ω) satisfies Pu = g(u) in Ω and u(x) = 0 in ω, then u vanishes in Ω.
Proof. The support of u is a closed set. Since F = supp(u) cannot be equal to Ω, let us show that F is open.
It will then follow that F = ∅. Assume that fr(F) = F \ F ◦ is not empty and chose x 1 ∈ fr(F). We set
A := Ω \ F. We recall that we denote by B(x, r) the Euclidean open ball with center x and radius r. There