of Carlos Kenig

The classical unique continuation theorem, which originates in the work of Carleman, in its simplest form is the following: Proposition: Assume that ∆u = V u in {|x| < 10}, with |u| ≤ C 0 and ||V || L ∞ ≤ M. If |u(x)| ≤ C N |x| N , for all N ≥ 0, then u ≡ 0. In order to establish this Proposition, Carleman developed a method, the " method of Carleman estimates " , which still permeates the subject. An example of such an estimate is the following one due to Hörmander (1983). Lemma: There exist C 1 , C 2 , C 3 , depending only on the dimension n, and an increasing function w(r), 0 < r < 10, so that 1 C 1 ≤ w(r) r ≤ C 1 and such that, for all f ∈ C ∞ 0 (B(0, 10)\{0}), α > C 2 , we have α 3 w (−1−2α) f 2 ≤ C 3 w (2−2α) |∆f | 2. I will give the proof of this Lemma later on, but let's illustrate Carleman's method by showing how it yields the Proposition. ∞ 0 (B(0, 10)), ϕ ≡ 1 on B(0, 2), 0 ≤ ϕ ≤ 1, ψ ∈ C ∞ (R n), ψ ≡ 1 for |x| ≥ 1, ψ ≡ 0 for |x| < 1/2. For > 0, small, let f (x) = ψ x ϕ(x)u(x), and apply the Lemma to f. We obtain (with η (x) = ψ x ϕ(x)): α 3 w −1−2α f 2 ≤ C 3 w 2−2α (∆f) 2 ≤ C 3 w 2−2α [η (x)∆u + 2∇η ∇u + ∆η u] 2. Note that ∇η = 1 ∇ψ x ϕ + ψ x ∇ϕ, while ∆η = 1 2 ∆ψ x ϕ + ψ x ∆ϕ + 2 ∇ψ x ∇ϕ. In order to control the term involving ∇u, we use the Caccioppoli inequality 1