Unique Continuation from Infinity for Waves, and Applications Arick Shao Queen Mary University of London Cardiff Analysis Seminar 23 January, 2017 Arick Shao (QMUL) Unique Continuation 1 / 30
Background on Wave Equations Wave Equations The (free) wave equation on flat spacetime: � φ := (− ∂ 2 φ : R t × R n t + ∆ x ) φ = 0, x → R . Generalisations: linear and nonlinear waves, systems, geometric waves. Examples of hyperbolic PDEs: Finite speed of propagation, dispersion, conservation, formation of singularities. Waves found in many equations of physics: Electromagnetism, gravitation, fluids. Arick Shao (QMUL) Unique Continuation 2 / 30
Background on Wave Equations Well-Posedness t Natural question—Cauchy problem: φ =? x φ = φ 0 t = 0 � φ = 0, φ | t = 0 = φ 0 , ∂ t φ | t = 0 = φ 1 . ∂ t φ = φ 1 φ =? The Cauchy problem is well-posed. The solution exists. 1 The solution is unique. 2 The solution is stable. (Depends continuously on initial data) 3 Given the current state of a system, we can predict the future. Arick Shao (QMUL) Unique Continuation 3 / 30
Background on Wave Equations Radiation Regular solutions of � φ = 0: Propagate at fixed, finite speed. t Decay in space and time at known rates. Can make sense of “asymptotics at infinity”: x Leading order coefficient: radiation field. Propagation of waves ( R 1 + 1 ). Physics: “what is seen by faraway observer” Electromagnetic, gravitational radiation. Arick Shao (QMUL) Unique Continuation 4 / 30
Background on Wave Equations Minkowski Geometry Theme: Geometric viewpoint for studying wave equations. Robust techniques applicable to many curved backgrounds. Applications to problems in relativity. Natural setting: Minkowski spacetime ( R 1 + n , m ) . Minkowski metric: m := − dt 2 + d ( x 1 ) 2 + · · · + d ( x n ) 2 . Setting of special relativity. � = m αβ ∇ αβ is m -trace of the Hessian. Natural second-order PDO associated with Minkowski geometry. Analogous to ∆ in Euclidean geometry. Arick Shao (QMUL) Unique Continuation 5 / 30
Background on Wave Equations Infinity T Infinity visualised via Penrose compactification. I + Conformal transformation m � → Ω 2 m . R 1 + n R ( R 1 + n , Ω 2 m ) isometrically embeds into relatively compact region in R × S n . I − Minkowski analogue of stereographic projection. R × S n , mod S n − 1 . T Infinity realised as boundary of shaded region. I + Future/past null infinity I ± : Null geodesics (bicharacteristics of � ) terminate here. R 1 + n R Radiation field manifested at I ± . I − Here, useful for pictures. Previous picture, projected. Arick Shao (QMUL) Unique Continuation 6 / 30
The Main Problem Main Questions Question (I) Can one reconstruct a wave from its values at infinity? In particular, are nonradiating waves trivial? Question (II) If so, can results be further generalised? Linear and nonlinear waves. Geometric waves (on curved backgrounds). Question (III) Can techniques behind (I) and (II) be applied to other problems? Arick Shao (QMUL) Unique Continuation 7 / 30
The Main Problem Scattering Theory There are classical scattering results for (I). (Friedlander) � φ = 0 on R 1 + n I + Initial data ( t = 0) ⇔ radiation field ( I + ). ⇒ Solve Cauchy problem. t = 0 ⇒ Solve backwards from radiation field. Generalisations: Red: solve forward from t = 0. Purple: solve backward from I + . Product manifolds R × X . Special nonlinear waves. Special black hole spacetimes. Arick Shao (QMUL) Unique Continuation 8 / 30
The Main Problem Beyond Scattering What about other wave equations? General linear and nonlinear waves, e.g., ( � + ∇ X + V ) φ = 0. Geometric waves on curved backgrounds, e.g., I + � g φ = g αβ ∇ 2 αβ φ = . . . . φ, ∇ φ = 0 φ = 0 ? Can we only impose data on part of I + and I − ? φ, ∇ φ = 0 Can waves be defined only locally near I ± ? I − An ill-posed question. These problems tend to be ill-posed: Cannot solve the wave equation. But, can ask if solutions are unique. Arick Shao (QMUL) Unique Continuation 9 / 30
The Main Problem The Result Near Infinity Theorem ( Alexakis–Schlue–S., 2015 ) I + Assume φ is a solution, near I ± , of φ, ∇ φ = 0 φ = 0 � φ + ∇ X φ + V φ = 0 , φ, ∇ φ = 0 where X, V decay sufficiently toward I ± . I − 2 + ε ) I ± , then φ = 0 in the If φ , ∇ φ vanish to ∞ -order on ( 1 interior near ( 1 2 + ε ) I ± . Uniqueness on shaded region, with data on ( 1 2 + ε ) I ± . Remark. The ∞ -order vanishing is optimal. Counterexamples if φ vanishes only to finite order. x r − 1 . n = 3: φ = ∇ k Arick Shao (QMUL) Unique Continuation 10 / 30
Geometric Unique Continuation Unique Continuation Formulate main question as unique continuation (UC) problem. Problem (Unique Continuation) Suppose: � g φ + ... = 0 Σ φ solves ( � g + ∇ X + V ) φ = 0 . φ, d φ vanish on a hypersurface Σ . φ = 0 ? φ, d φ = 0 Must φ vanish on one side of Σ ? Unique continuation problem. In particular, we are interested in Σ ⊆ I ± . Arick Shao (QMUL) Unique Continuation 11 / 30
Geometric Unique Continuation The Classical Theory Ancient theory: analytic PDE, noncharacteristic Σ . (Cauchy–Kovalevskaya) Existence, uniqueness of analytic solutions. (Holmgren, F. John) Solution unique even in nonanalytic classes. Classical theory for non-analytic equations (H¨ ormander): Crucial point: pseudoconvexity of Σ . Pseudoconvexity ⇒ Carleman estimates ⇒ UC Classical results are purely local. Applies to small neighbourhoods of P ∈ Σ . Arick Shao (QMUL) Unique Continuation 12 / 30
Geometric Unique Continuation The Geometric Approach Σ := { f = 0 } is pseudoconvex (wrt � g and f ) ⇔ null geodesic Σ ∇ 2 f ( X , X ) < 0 on Σ , if g ( X , X ) = Xf = 0. f > 0 P f < 0 − f convex in tangent null (bicharacteristic) directions. Null curve hitting Σ tangentially lies in { f < 0 } nearby. Σ pseudoconvex at P . Relativity: bending of light. Note. Pseudoconvexity is conformally invariant. Sensible to take Σ ⊆ I ± . Arick Shao (QMUL) Unique Continuation 13 / 30
Geometric Unique Continuation Zero Pseudoconvexity We next consider the (non-classical) borderline case. Σ is zero pseudoconvex ⇔ Σ is ruled by null geodesics. Also conformally invariant. Bad news: I ± is zero pseudoconvex, not pseudoconvex. Possible loss of local UC in zero pseudoconvex settings: (Alinhac–Baouendi) Counterexample to local UC when Σ = { x n = 0 } ⊆ R 1 + n . (Kenig–Ruiz–Sogge) Global UC from all of Σ . Main result: Semi-global UC (from “large enough” part of I ± ). Arick Shao (QMUL) Unique Continuation 14 / 30
Uniqueness Near Infinity Finding Pseudoconvexity We now return to the main problem. ι + Q1. Can we find pseudoconvexity near I ± ? I + Consider hyperboloids in R 1 + n ι 0 Blue level sets of r 2 − t 2 . I − These are only zero pseudoconvex. ι − Idea. Take instead ( 1 2 + ε ) I ± ”. Blue: level hyperboloids of f . Consider red “warped hyperboloids”. Red: pseudoconvex hyperboloids. These are (inward) pseudoconvex. Arick Shao (QMUL) Unique Continuation 15 / 30
Uniqueness Near Infinity Strong Unique Continuation Q2. Where do we see the ∞ -order vanishing? By another conformal transformation: I + 2 + ε ) I ± ⇔ UC from cone r 2 − t 2 = 0. UC from ( 1 I ¯ (But with “warped” geometry.) ι 0 f = c − UV = c Strong UC for elliptic equations: I − UC from a single point (say r 2 = 0). Warped inversion of Minkowski. Requires ∞ -order vanishing (in r ). (Figure by V. Schlue.) UC from r 2 − t 2 = 0: hyperbolic analogue of strong UC. Arick Shao (QMUL) Unique Continuation 16 / 30
Uniqueness Near Infinity Carleman Estimates Q1 + Q2 + hard work ⇒ Carleman estimates. Weighted spacetime integral inequalities. Standard tool for proving UC. Technical difficulties due to degenerate pseudoconvexity. Carleman estimates + standard argument ⇒ main UC result. Remark. Carleman estimates useful in many other PDE problems. Inverse problems. Controllability of PDEs. Arick Shao (QMUL) Unique Continuation 17 / 30
Uniqueness Near Infinity Geometric Robustness Carleman estimate proved using geometric computations. Main result is geometrically robust—extend to many curved backgrounds. Asymptotically flat spacetimes: those with “similar structure of infinity”. Theorem ( Alexakis–Schlue–S., 2015 ) I + The main result extends to a large class of (both stationary and dynamic) asymptotically flat spacetimes, including: φ, ∇ φ = 0 Perturbations of Minkowski spacetimes. 1 I − Schwarzschild and Kerr spacetimes, and perturbations. 2 For (2), result can be localised near ε I ± . UC from ε I ± to shaded region. Arick Shao (QMUL) Unique Continuation 18 / 30
Global Uniqueness Toward Finite-Order Vanishing Let us return to Minkowski spacetime ( R 1 + n , m ) Question. Can we remove ∞ -order vanishing condition? Recall. Counterexamples to finite-order vanishing: x r − 1 ∇ k ( n = 3). These counterexamples blow up at r = 0. Same property holds for other dimensions. Idea. Impose global regularity for φ . Eliminates above counterexamples. Arick Shao (QMUL) Unique Continuation 19 / 30
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