Unique Continuation for Waves, Carleman Estimates, and Applications - PowerPoint PPT Presentation

. . . . . . . . . . . . . . . Unique Continuation for Waves, Carleman Estimates, and Applications Arick Shao Queen Mary University of London Leeds Analysis and Applications Seminar 14 February, 2018 Arick Shao (QMUL) Unique Continuation . . . . . . . . . . . . . . . . . . . . . . . . . 1 / 36

. Recent unique continuation (UC) results for wave equations. . . . . . . . . . Outline 1 UC “from infjnity”. . 2 Theory of UC. Why is “classical” theory not enough? 3 Some ideas behind Carleman estimates. Main analysis tool for UC. 4 (Other) Applications of Carleman estimates Focus on control theory. Arick Shao (QMUL) Unique Continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 / 36

. . . . . . . . . . . . . . . . Unique Continuation from Infjnity Section 1 Unique Continuation from Infjnity Arick Shao (QMUL) Unique Continuation . . . . . . . . . . . . . . . . . . . . . . . . 3 / 36

. . . . . . . . . . . . . . Unique Continuation from Infjnity Wave Equations Consider the wave equation: Generalisations: linear/nonlinear waves, systems, geometric waves. Physics: Maxwell equations, Yang-Mills equations, Einstein equations, fmuids Initial value problem: Solution “depends continuously on” initial data. Arick Shao (QMUL) Unique Continuation . . . . . . . . . . . . . . . . . . . . . . . . . . 4 / 36 □ φ := (− ∂ 2 t + ∆ x ) φ = 0, φ : R t × R n x → R . □ φ = F ( t , x , φ, D φ ) , φ | t = 0 = φ 0 , ∂ t φ | t = 0 = φ 1 . In general, ∃ ! solution for “nice” initial data ( φ 0 , φ 1 ) .

. Unique Continuation from Infjnity . . . . . . . . . . Radiation . x t Propagate at fjxed, fjnite speed. Decay in space and time at known rates. Can make sense of “asymptotics at infjnity”: Leading order coeffjcient: radiation fjeld. Question (UC from infjnity) Are solutions of wave equations determined by its “data at infjnity”? Arick Shao (QMUL) Unique Continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 / 36 Regular solutions of □ φ = 0: Propagation of waves ( R 1 + 1 ).

. . . . . . . . . . . . . . . Unique Continuation from Infjnity Minkowski Geometry Theme: Geometric viewpoint for studying wave equations. Robust techniques applicable to many curved backgrounds. Applications to problems in relativity. Setting of special relativity. Arick Shao (QMUL) Unique Continuation . . . . . . . . . . . . . . . . . . . . . . . . . 6 / 36 Natural setting: Minkowski spacetime ( R 1 + n , m ) . Minkowski metric: m := − dt 2 + d ( x 1 ) 2 + · · · + d ( x n ) 2 . □ = m αβ ∇ αβ : natural second-order PDO in Minkowski geometry. Analogue of ∆ in Euclidean geometry.

. . . . . . . . . . . . Unique Continuation from Infjnity . Infjnity R T R T Previous picture, projected. Infjnity visualised via Penrose compactifjcation. Infjnity realised as boundary of shaded region. For this talk, useful for drawing pictures. Arick Shao (QMUL) Unique Continuation . . . . . . . . . . . . . . . 7 / 36 . . . . . . . . . . . . I + Conformal transformation m � → Ω 2 m . R 1 + n ( R 1 + n , Ω 2 m ) isometrically embeds into relatively I − compact region in R × S n . R × S n , mod S n − 1 . Future/past null infjnity I ± : Null geodesics I + (bicharacteristics of □ ) terminate here. R 1 + n Radiation fjeld manifested at I ± . I −

. . . . . . . . . . . . . . . Unique Continuation from Infjnity Main Questions Question (UC from infjnity) For linear waves: Are nonradiating waves trivial? Arick Shao (QMUL) Unique Continuation . . . . . . . . . . . . . . . . . . . . . . . . . 8 / 36 Does φ on some part of I ± determine φ inside? General linear/nonlinear waves, e.g., ( □ + ∇ X + V ) φ = 0 ? Geometric waves on curved backgrounds: □ g φ = g αβ ∇ 2 αβ φ = . . . ? I + φ = 0 If φ = 0 on some part of I ± , then is φ = 0 inside? φ = 0 ? φ = 0 I −

. . . . . . . . . . . . . . Unique Continuation from Infjnity Scattering Results (Friedlander) Isometry between initial data (at Various generalisations: special black hole spacetimes. However, we are more interested in: Ill-posed settings: cannot solve the wave equation. Other linear and geometric waves. Arick Shao (QMUL) Unique Continuation . . . . . . . . . . . . . . . . . . . . . . 9 / 36 . . . . t = 0) and radiation fjeld (at I + ). I + Applies only to free waves ( □ φ = 0). t = 0 Red: Solve forward from t = 0. Product manifolds R × X , special nonlinear waves, Blue: Solve backward from I + .

. . . . . . . . . . . . . . . . Unique Continuation from Infjnity Result Near Infjnity Theorem ( Alexakis–Schlue–S., 2015 ) Arick Shao (QMUL) Unique Continuation . . . . . . . . . . . . . . . . . . . 10 / 36 . . . . . Assume φ is a solution, near I ± , of I + φ, ∇ φ = 0 □ φ + ∇ X φ + V φ = 0 , φ = 0 where X, V decay suffjciently toward I ± . φ, ∇ φ = 0 2 + ε ) I ± , then φ = 0 in the If φ , ∇ φ vanish to ∞ -order on ( 1 I − 2 + ε ) I ± . interior near ( 1 Remark. The ∞ -order vanishing is optimal. Counterexamples if φ vanishes only to fjnite order. On R 1 + n ( n > 2), can take φ = ∇ k x r −( n − 2 ) .

. Geometric Robustness . . . . . . . . . . Unique Continuation from Infjnity Question . Can UC result be extended to curved backgrounds? Asymptotically fmat spacetimes: those with “similar structure of infjnity”. Theorem ( Alexakis–Schlue–S., 2015 ) The main result extends to a large class of (both stationary and dynamic) asymptotically fmat spacetimes, including: 1 Perturbations of Minkowski spacetimes. 2 Schwarzschild and Kerr spacetimes, and perturbations. Arick Shao (QMUL) Unique Continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 / 36 I + φ, ∇ φ = 0 I − For (2), result can be localised near ε I ± .

. . . . . . . . . . . . . . . . Unique Continuation from Infjnity Finite-Order Vanishing? Question Arick Shao (QMUL) Unique Continuation . . . . . . . . . . . . . . . . . . . 12 / 36 . . . . . On Minkowski spacetime ( R 1 + n , m ) : Can ∞ -order vanishing condition be somehow removed? x r −( n − 2 ) . Recall. Counterexamples ∇ k Note that these blow up at r = 0. Idea. Impose global regularity for φ , up to r = 0.

. . . . . . . . . . . . . . . . Unique Continuation from Infjnity The Global Result Theorem ( Alexakis–S., 2015 ) Otherwise, there are counterexamples. Arick Shao (QMUL) Unique Continuation . . . . . . . . . . . . . . . . . . . 13 / 36 . . . . . I + Assume φ is a regular solution in D of φ, ∇ φ = 0 D □ φ + V φ = 0 , ∥ V ∥ L ∞ ≤ C, φ = 0 φ, ∇ φ = 0 where V also decays toward I ± as before. I − 2 I ± , with A 0 depending If φ , ∇ φ vanish to order A > A 0 at 1 on C, then φ = 0 everywhere on D (and hence R 1 + n ). Remark. The L ∞ -assumption on V is necessary.

. . . . . . . . . . . . . . . Unique Continuation from Infjnity The Global Nonlinear Theorem Question Can some special wave equations be better behaved? Theorem ( Alexakis–S., 2015 ) Arick Shao (QMUL) Unique Continuation . . . . . . . . . . . . . . . . . . . . 14 / 36 . . . . . Suppose φ is a regular solution in D of □ φ + V | φ | p − 1 φ = 0 , p ≥ 1 , where V satisfjes a monotonicity property (depending on p). 2 I ± for any δ > 0 , then φ = 0 on D . If φ , ∇ φ vanishes to order δ at 1 Idea. Estimates not for □ , but for □ V , p φ := □ φ + V · | φ | p − 1 φ .

. . . . . . . . . . . . . . . . Unique Continuation Theory Section 2 Unique Continuation Theory Arick Shao (QMUL) Unique Continuation . . . . . . . . . . . . . . . . . . . . . . . . 15 / 36

. . . . . . . . . . . . . . . Unique Continuation Theory Unique Continuation Unique continuation (UC): classical problem in PDEs. When we cannot solve a PDE, we can still ask if solutions are unique. Problem (Unique Continuation) Suppose: Arick Shao (QMUL) Unique Continuation . . . . . . . . . . . . . . . . . . . . 16 / 36 . . . . . □ g φ + ... = 0 φ solves ( □ g + ∇ X + V ) φ = 0 . Σ φ, ∇ φ vanish on a hypersurface Σ . φ = 0 ? φ, ∇ φ = 0 Must φ vanish on one side of Σ ? In particular, we are interested in Σ ⊆ I ± .

. . . . . . . . . . . . . . . Unique Continuation Theory The Classical Theory (Cauchy–Kovalevskaya) Existence, uniqueness of analytic solutions. (Holmgren, F. John) Solution unique even in nonanalytic classes. Classical theory for non-analytic equations (Calderón, Hörmander): Remark. Classical UC results are purely local. Arick Shao (QMUL) Unique Continuation . . . . . . . . . . . . . . . . . . . . . . . . . 17 / 36 Ancient theory: analytic PDE, noncharacteristic Σ . Crucial point: pseudoconvexity of Σ . Σ pseudoconvex ⇒ Carleman estimates ⇒ UC from Σ . (Alinhac–Baouendi) Σ not pseudoconvex ⇒ ∃ X , V with counterexamples. UC from a small neighbourhood of P ∈ Σ .