Control from an Interior Hypersurface

Transcription

1 Control from an Interior Hypersurface Matthieu Léautaud École Polytechnique Joint with Jeffrey Galkowski Murramarang, microlocal analysis on the beach March,

2 Outline General questions Eigenfunctions A very weak lower bound A GCC-type result Wave equation Two results Solution to the wave equation? Controllability/observability for the wave equation

3 Setting: Questions (M, g) be a compact n dimensional Riemannian manifold, possibly with boundary M, g the (non-positive) Laplace-Beltrami operator on M, Σ is an interior hypersurface (dim Σ = n 1, Σ M =, global normal vector field ν). General questions: What is the mass / energy left by solutions to g λ 2 or 2 t g or t g on the hypersurface Σ? Can we observe solutions to g λ 2 or 2 t g or t g on the hypersurface Σ? Can we control solutions to 2 t g or t g from the hypersurface Σ? Usual situations: open set ω M, or open subset of the boundary Γ M.

4 First Break

5 Eigenfunctions: some classical results ( g λ 2 )φ = 0, φ M = 0. (Eig) ω M, open Study: φ L 2 (ω), Weak lower bound (ω ): φ L 2 (ω) ce λ/c φ L 2 (M), [Donnelly-Fefferman 88]

6 Eigenfunctions: some classical results ( g λ 2 )φ = 0, φ M = 0. (Eig) ω M, open Study: φ L 2 (ω), Weak lower bound (ω ): φ L 2 (ω) ce λ/c φ L 2 (M), [Donnelly-Fefferman 88] Γ M, open Study: nφ Γ L 2 (Γ) Weak lower bound (Γ ): nφ Γ L 2 (Γ) ce λ/c φ L 2 (M), [Lebeau-Robbiano 95]

7 Eigenfunctions: some classical results ( g λ 2 )φ = 0, φ M = 0. (Eig) ω M, open Study: φ L 2 (ω), Weak lower bound (ω ): φ L 2 (ω) ce λ/c φ L 2 (M), [Donnelly-Fefferman 88] Γ M, open Study: nφ Γ L 2 (Γ) Weak lower bound (Γ ): nφ Γ L 2 (Γ) ce λ/c φ L 2 (M), [Lebeau-Robbiano 95] Tunneling estimates for eigenfunctions: No assumption = weak lower bound (optimal in general)!

8 Eigenfunctions: some classical results ( g λ 2 )φ = 0, φ M = 0. (Eig) ω M, open Study: φ L 2 (ω), Weak lower bound (ω ): φ L 2 (ω) ce λ/c φ L 2 (M), [Donnelly-Fefferman 88] Γ M, open Study: nφ Γ L 2 (Γ) Weak lower bound (Γ ): nφ Γ L 2 (Γ) ce λ/c φ L 2 (M), [Lebeau-Robbiano 95] Tunneling estimates for eigenfunctions: No assumption = weak lower bound (optimal in general)! Improve under a geometric condition on ω?

9 Eigenfunctions: some classical results ( g λ 2 )φ = 0, φ M = 0. (Eig) Assumption (GCC): there exists L > 0 such that all (generalized) geodesics of length L intersect ω. Then φ L 2 (ω) c φ L 2 (M), [Bardos-Lebeau-Rauch 92] Sometimes but not always necessary for eigenfunctions (always necessary for o(λ 1 )-quasimodes)

10 Eigenfunctions: some classical results ( g λ 2 )φ = 0, φ M = 0. (Eig) Assumption (GCC): there exists L > 0 such that all (generalized) geodesics of length L intersect ω. Then φ L 2 (ω) c φ L 2 (M), [Bardos-Lebeau-Rauch 92] Sometimes but not always necessary for eigenfunctions (always necessary for o(λ 1 )-quasimodes) Assumption (GCC): there exists L > 0 such that all (generalized) geodesics of length L intersect Γ (at a non-diffractive point). Then λ 1 nφ Γ L 2 (Γ) c φ L 2 (M), [Bardos-Lebeau-Rauch 92]

11 Eigenfunctions: a very weak lower bound ( g λ 2 )φ = 0, φ M = 0. (Eig) Theorem Assume M is connected and Int(Σ) is nonempty. Then there exists c > 0 so that for all λ 0 and φ L 2 (M) solutions to (Eig), we have φ Σ L 2 (Σ) + νφ Σ L 2 (Σ) ce λ/c φ L 2 (M). Recall [Donnelly-Fefferman 88], [Lebeau-Robbiano 95]: φ L 2 (ω) ce λ/c φ L 2 (M), nφ Γ L 2 (Γ) ce λ/c φ L 2 (M).

12 Eigenfunctions: a very weak lower bound Proposition (Optimality in general) Consider the manifold ( g λ 2 )φ = 0, φ M = 0. (Eig) M = [ π, π] z T 1 θ,, g(z, θ) = dz 2 + R(z) 2 dθ 2. Assume that R is even and has two bumps. Let Σ = {z = 0} T 1 M. Then, there exist C, c > 0 and sequences λ e / o j + and φ e / o j L 2 (M) such that ( (λ e / o j ) 2 )φ e / o j = 0, φ e / o j L 2 (M) = 1, φ e / o j M = 0, with νφ e j Σ = 0, φ e j Σ L 2 (Σ) Ce cλe j, φ o j Σ = 0, νφ o j Σ L 2 (Σ) Ce cλo j.

13 Eigenfunctions: a very weak lower bound Open question: can we remove one of the two terms? Conjecture (?) Assume Σ has positive definite second fundamental form. Then there exists C, c, λ 0 > 0 so that for all (λ, φ) [λ 0, ) L 2 (M) satisfying (Eig), we have φ Σ L 2 (Σ) Ce cλ φ L 2 (M), and λ 1 νφ Σ L 2 (Σ) Ce cλ φ L 2 (M). This would have applications (cf Toth-Zelditch 17 for nodal domains)

14 Eigenfunctions: a GCC-type result An improved lower bound under a geometric condition. Assumption (T GCC) ( g λ 2 )φ = 0, φ M = 0. (Eig) There is L > 0 s.t. all (generalized) geodesics of length L cross Σ transversally. Theorem Assume (T GCC) then there is c > 0 so that for all λ 0 and φ L 2 (M) solutions to (Eig), we have φ Σ L 2 (Σ) + λ 1 νφ Σ L 2 (Σ) c φ L 2 (M). (1) [BLR 92]: Under GCC φ L 2 (ω) c φ L 2 (M), λ 1 nφ Γ L 2 (Γ) c φ L 2 (M).

15 Eigenfunctions: a GCC-type result Removing T in (T GCC)? Proposition Assume M = S 2 and Σ is a great circle. Then there exists a sequence (λ j, φ j ) satisfying ( g λ 2 j )φ j = 0 together with λ j + and φ j Σ = 0, j νφ j Σ L 2 (Σ) λ 1/4 φ j L 2 (M). λ 1 j Beware that φ Σ L 2 (Σ) might be unbounded as λ + : take M = S 2 and Σ is a great circle. There is another sequence s.t. φ j Σ L 2 (Σ) cλ 1/4 j different regularity/growth on the glancing set! [Galkowski 16]

16 Eigenfunctions: a GCC-type result Open question: can we remove one of the two terms? Conjecture (?) Assume further that Σ has positive definite second fundamental form. Then there exists C, c, λ 0 > 0 so that for all (λ, φ) [λ 0, ) L 2 (M) satisfying (Eig), we have φ L 2 (M) C φ Σ L 2 (Σ), and φ L 2 (M) C λ 1 νφ Σ L 2 (Σ).

17 Resolvent estimates/quasimodes For all λ 0 and all u H 2 (M) H 1 0 (M) we have u L 2 (M) Ce cλ( u Σ L 2 (Σ) + λ 1 νu Σ L 2 (Σ) + ( g λ 2 )u L 2 (M)). Under (T GCC) u L 2 (M) C ( u Σ L 2 (Σ) + λ 1 νu Σ L 2 (Σ) + λ 1 ( g λ 2 )u L 2 (M)).

18 Related questions/works General upper bounds: [Tataru 98], L p estimates [Burq-Gérard-Tzvetkov 07] [Tacy 10-14] [Christianson-Hassell-Toth 15] Near the glancing set [Galkowski 16] Lower bounds on M = T 2, T 3 : Bourgain and Rudnick [Bourgain-Rudnick 09-12] If Σ is a real analytic hypersurface with nonvanishing curvatures, then C 1 φ L 2 (M) φ Σ L 2 (Σ) C φ L 2 (M) Density one subsequences of eigenfunctions equidistribute: Quantum Ergodic Restriction [Toth-Zelditch ] [Dyatlov-Zworski 12], Torus [Hezari-Rivière 18]

19 Second break

20 The wave equation controlled from Σ Goal: control the wave equation from Σ: v = f 0δ Σ + f 1δ Σ on (0, T ) Int(M), v = 0 on (0, T ) M, (v, tv) t=0 = (v 0, v 1) in Int(M). (Wave) where f 0δ Σ, ϕ = Σ f 0ϕdσ, f 1δ Σ, ϕ = f 1 νϕdσ. Σ More precisely: Given (v 0, v 1) and T > 0, find (f 0, f 1) such that (v, tv) t=t = (0, 0). (equivalent to control to any target) Question 0: what is a solution? well-posedness?

21 The wave equation controlled from Σ Goal: control the wave equation from Σ: v = f 0δ Σ + f 1δ Σ on (0, T ) Int(M), v = 0 on (0, T ) M, (v, tv) t=0 = (v 0, v 1) in Int(M). (Wave) where f 0δ Σ, ϕ = Σ f 0ϕdσ, f 1δ Σ, ϕ = f 1 νϕdσ. Σ More precisely: Given (v 0, v 1) and T > 0, find (f 0, f 1) such that (v, tv) t=t = (0, 0). (equivalent to control to any target) Question 0: what is a solution? well-posedness? [Bardos Lebeau Rauch 92]: v = 1 ωf, v = 0, v M = 1 Γ f

22 The wave equation controlled from Σ General spirit [J.L. Lions 70-90]: duality Well-posedness regularity estimate for the free equation (direct inequality) Controllability observability estimate for the free equation (reverse inequality)

23 The wave equation controlled from Σ Normal coordinates: x = (x 1, x ), Σ = {x 1 = 0}, x 1 = dist g (x, Σ), ν = x1, σ( g ) = ξ1 2 + r(x 1, x, ξ ), (think r(x 1, x, ξ ) = ξ 2 ) σ( ) = τ 2 + σ( g ) = τ 2 + ξ1 2 + r(x 1, x, ξ ).

24 The wave equation controlled from Σ Normal coordinates: x = (x 1, x ), Σ = {x 1 = 0}, x 1 = dist g (x, Σ), ν = x1, σ( g ) = ξ1 2 + r(x 1, x, ξ ), (think r(x 1, x, ξ ) = ξ 2 ) σ( ) = τ 2 + σ( g ) = τ 2 + ξ1 2 + r(x 1, x, ξ ). Conical subsets of TR Int(Σ)(R M) \ 0 E = {(t, 0, x, τ, ξ 1, ξ ) ξ1 2 + r(0, x, ξ ) τ 2 }, T = {(t, 0, x, τ, ξ 1, ξ ) ξ1 2 + r(0, x, ξ ) = τ 2, ξ1 2 > 0}, G = {(t, 0, x, τ, ξ 1, ξ ) ξ1 2 + r(0, x, ξ ) = τ 2, ξ1 2 = 0}.

25 The wave equation controlled from Σ Normal coordinates: x = (x 1, x ), Σ = {x 1 = 0}, x 1 = dist g (x, Σ), ν = x1, σ( g ) = ξ1 2 + r(x 1, x, ξ ), (think r(x 1, x, ξ ) = ξ 2 ) σ( ) = τ 2 + σ( g ) = τ 2 + ξ1 2 + r(x 1, x, ξ ). Conical subsets of TR Int(Σ)(R M) \ 0 E = {(t, 0, x, τ, ξ 1, ξ ) ξ1 2 + r(0, x, ξ ) τ 2 }, T = {(t, 0, x, τ, ξ 1, ξ ) ξ1 2 + r(0, x, ξ ) = τ 2, ξ1 2 > 0}, G = {(t, 0, x, τ, ξ 1, ξ ) ξ1 2 + r(0, x, ξ ) = τ 2, ξ1 2 = 0}. Conical subsets of T (R Int(Σ)) \ 0 E Σ := {(t, x, τ, ξ ) r(0, x, ξ ) > τ 2 }, T Σ := {(t, x, τ, ξ ) r(0, x, ξ ) < τ 2 }, G Σ := {(t, x, τ, ξ ) r(0, x, ξ ) = τ 2 }.

26 The wave equation controlled from Σ: well-posedness v = f 0δ Σ + f 1δ Σ v = 0 (v, tv) t=0 = (v 0, v 1) on R + Int(M), on R + M, in Int(M). (Wave) Theorem For all (v 0, v 1) L 2 (M) H 1 (M) and for all f 0 Hcomp(R 1 + Int(Σ)) and f 1 L 2 comp(r + Int(Σ)) such that WF 2 1 (f0), WF 2 1 (f1) G Σ =, there exists a unique v L 2 loc(r +; L 2 (M)) solution of (Wave). Well-posedness OK if (f 0, f 1) are H 1 (R Σ) L 2 (R Σ) overall H 1 2 H 1 2 microlocally near the glancing set G Σ Poor regularity in time of the solution!

27 The wave equation controlled from Σ: control! Definition (Σ, T ) satisfies (T GCC) if every (generalized) bicharacteristic of intersects T (0,T ) Int(Σ)(R M) \ G. Theorem Assume (Σ, T ) satisfies (T GCC). Then for any (v 0, v 1) L 2 (M) H 1 (M) there exist (f 0, f 1) Hcomp((0, 1 T ) Int(Σ)) L 2 comp((0, T ) Int(Σ)) with WF(f 0), WF(f 1) (G Σ E Σ ) =, so that the solution to (Wave) has v 0 for t T.

28 Solution of the wave equation controlled from Σ Assume v, u, F solve (in a reasonable sense): v = f 0δ Σ + f 1δ Σ in D ((0, T ) Int(M)), and u = F. Multiply by u and integrate by parts [ ] T ( tv, u) L 2 (M) (v, tu) L 2 (M) +(v, F ) 0 L 2 ((0,T ) M) = (0,T ) Σ (f 0u Σ f 1 νu Σ ) dtdσ.

29 Solution of the wave equation controlled from Σ Assume v, u, F solve (in a reasonable sense): v = f 0δ Σ + f 1δ Σ in D ((0, T ) Int(M)), and u = F. Multiply by u and integrate by parts [ ] T ( tv, u) L 2 (M) (v, tu) L 2 (M) +(v, F ) 0 L 2 ((0,T ) M) = For this to make sense as a definition (in a weak sense): (0,T ) Σ take F s as test functions (Riesz Representation theorem) the regularity f 0 should fit that of u Σ (u sol free wave) F L 2 ((0, T ) M), (u, tu) H 1 0 L 2 will yield a solution v L 2 ((0, T ) M) for data (v, tv) t=0 L 2 H 1 (f 0u Σ f 1 νu Σ ) dtdσ. Main question: what is the regularity of u Σ, νu Σ (for u sol of u = F, F L 2 ((0, T ) M), (u, tu) H 1 0 L 2? Then, take this as a definition of a solution!

30 Solution of the wave equation controlled from Σ Question: regularity of u Σ, νu Σ for u sol of u = F, F L 2 ((0, T ) M), (u, tu) H 1 0 L 2?

31 Solution of the wave equation controlled from Σ Question: regularity of u Σ, νu Σ for u sol of u = F, F L 2 ((0, T ) M), (u, tu) H 1 0 L 2? u H 1 overall (Cauchy problem)

32 Solution of the wave equation controlled from Σ Question: regularity of u Σ, νu Σ for u sol of u = F, F L 2 ((0, T ) M), (u, tu) H0 1 L 2? u H 1 overall (Cauchy problem) u Σ H 1/2 ε, νu Σ H 1/2 ε overall (usual trace theorem)

33 Solution of the wave equation controlled from Σ Question: regularity of u Σ, νu Σ for u sol of u = F, F L 2 ((0, T ) M), (u, tu) H0 1 L 2? u H 1 overall (Cauchy problem) u Σ H 1/2 ε, νu Σ H 1/2 ε overall (usual trace theorem) u Σ H 1/2, νu Σ H 1/2 overall ( u = F, elliptic on the conormal)

34 Solution of the wave equation controlled from Σ Question: regularity of u Σ, νu Σ for u sol of u = F, F L 2 ((0, T ) M), (u, tu) H0 1 L 2? u H 1 overall (Cauchy problem) u Σ H 1/2 ε, νu Σ H 1/2 ε overall (usual trace theorem) u Σ H 1/2, νu Σ H 1/2 overall ( u = F, elliptic on the conormal) Near E Σ (elliptic region) u Σ H 3/2, νu Σ H 1/2

35 Solution of the wave equation controlled from Σ Question: regularity of u Σ, νu Σ for u sol of u = F, F L 2 ((0, T ) M), (u, tu) H 1 0 L 2? u H 1 overall (Cauchy problem) u Σ H 1/2 ε, νu Σ H 1/2 ε overall (usual trace theorem) u Σ H 1/2, νu Σ H 1/2 overall ( u = F, elliptic on the conormal) Near E Σ (elliptic region) u Σ H 3/2, νu Σ H 1/2 Near T Σ (hyperbolic/transversal region) u Σ H 1, νu Σ L 2 (Cauchy problem w.r.t. Σ)

36 Solution of the wave equation controlled from Σ Question: regularity of u Σ, νu Σ for u sol of u = F, F L 2 ((0, T ) M), (u, tu) H 1 0 L 2? u H 1 overall (Cauchy problem) u Σ H 1/2 ε, νu Σ H 1/2 ε overall (usual trace theorem) u Σ H 1/2, νu Σ H 1/2 overall ( u = F, elliptic on the conormal) Near E Σ (elliptic region) u Σ H 3/2, νu Σ H 1/2 Near T Σ (hyperbolic/transversal region) u Σ H 1, νu Σ L 2 (Cauchy problem w.r.t. Σ) Conclusion: u Σ H 1/2, νu Σ H 1/2 overall u Σ H 1, νu Σ L 2 away from G Σ

37 Solution of the wave equation controlled from Σ Question: regularity of u Σ, νu Σ for u sol of Conclusion: u = F, F L 2 ((0, T ) M), (u, tu) H 1 0 L 2? u Σ H 1, νu Σ L 2 away from G Σ u Σ H 1/2, νu Σ H 1/2 near G Σ (overall) In view of [ ] T ( tv, u) L 2 (M) (v, tu) L 2 (M) +(v, F ) 0 L 2 ((0,T ) M) = we should take: f 0 H 1, f 1 L 2 away from G Σ f 0 H 1/2, f 1 H 1/2 near G Σ (0,T ) Σ (f 0u Σ f 1 νu Σ ) dtdσ, Remark: any improvement of restriction estimates near G Σ lowers the regularity requirements for f 0, f 1 for well-posedness

38 The wave equation controlled from Σ: controllability Controllability an observability estimate (spirit: A is onto v C A v ) (Usual case: Controllability from ω (u 0, u 1) H 1 L 2 C u L 2 ((0,T );H 1 (ω)) if u = 0 )

39 The wave equation controlled from Σ: controllability Controllability an observability estimate (spirit: A is onto v C A v ) (Usual case: Controllability from ω (u 0, u 1) H 1 L 2 C u L 2 ((0,T );H 1 (ω)) if u = 0 ) Theorem (Observability estimate) Under Assumption (T GCC), there exists δ > 0 s.t. with A δ Ψ 0 phg((0, T ) Int(Σ)) with principal symbol as on the picture; ϕ δ Cc ((0, T ) Int(Σ)) with ϕ δ 1 on [δ, T δ] Σ δ we have, with u = F, (u, tu) t=0 = (u 0, u 1), c N (u 0, u 1) 2 H 1 L 2 A δ( νu Σ ) 2 L 2 (R Σ) + A δ (u Σ ) 2 H 1 (R Σ) + F 2 L 2 ((0,T ) M)

40 The wave equation controlled from Σ: controllability Controllability an observability estimate (spirit: A is onto v C A v ) (Usual case: Controllability from ω (u 0, u 1) H 1 L 2 C u L 2 ((0,T );H 1 (ω)) if u = 0 ) Theorem (Observability estimate) Under Assumption (T GCC), there exists δ > 0 s.t. with A δ Ψ 0 phg((0, T ) Int(Σ)) with principal symbol as on the picture; ϕ δ Cc ((0, T ) Int(Σ)) with ϕ δ 1 on [δ, T δ] Σ δ we have, with u = F, (u, tu) t=0 = (u 0, u 1), c N (u 0, u 1) 2 H 1 L 2 A δ( νu Σ ) 2 L 2 (R Σ) + A δ (u Σ ) 2 H 1 (R Σ) + F 2 L 2 ((0,T ) M) + ϕ δ νu Σ 2 H N (R Σ) + ϕ δu Σ 2 H N (R Σ)

41 The wave equation controlled from Σ: observability Goal: Prove c N (u 0, u 1) 2 H 1 L 2 A δ( νu Σ ) 2 L 2 (R Σ) + A δ (u Σ ) 2 H 1 (R Σ) + F 2 L 2 ((0,T ) M) + ϕ δ νu Σ 2 H N (R Σ) + ϕ δu Σ 2 H N (R Σ) Three steps: (T GCC) = (T GCC) δ uniformly away from G High frequency estimate using (T GCC) δ Low frequencies: unique continuation from Σ

42 The wave equation controlled from Σ: observability at high frequency Near T, we have τ 2 > r(x, ξ ): ( σ( ) = ξ1 2 + r(x, ξ ) τ 2 = ξ 1 ) ( τ 2 r(x, ξ ) ξ 1 + ) τ 2 r(x, ξ ) (D x1 Op(λ))(D x1 + Op(λ)) with λ = τ 2 r(x, ξ )

43 The wave equation controlled from Σ: observability at high frequency Near T, we have τ 2 > r(x, ξ ): ( σ( ) = ξ1 2 + r(x, ξ ) τ 2 = ξ 1 ) ( τ 2 r(x, ξ ) ξ 1 + ) τ 2 r(x, ξ ) (D x1 Op(λ))(D x1 + Op(λ)) with λ = τ 2 r(x, ξ ) Then solve the Cauchy problem in the x 1-variable: implies (D x1 ± Op(λ))w = f, w(x 1, ) L 2 (( ε,ε) R d ) C( w(0, ) L 2 (R d ) + f L 2 (( ε,ε) R d )) = boundary data u Σ, νu Σ dominate u near T.

44 The wave equation controlled from Σ: observability at high frequency Near T, we have τ 2 > r(x, ξ ): ( σ( ) = ξ1 2 + r(x, ξ ) τ 2 = ξ 1 ) ( τ 2 r(x, ξ ) ξ 1 + ) τ 2 r(x, ξ ) (D x1 Op(λ))(D x1 + Op(λ)) with λ = τ 2 r(x, ξ ) Then solve the Cauchy problem in the x 1-variable: implies (D x1 ± Op(λ))w = f, w(x 1, ) L 2 (( ε,ε) R d ) C( w(0, ) L 2 (R d ) + f L 2 (( ε,ε) R d )) = boundary data u Σ, νu Σ dominate u near T. Then: propagation argument: u near T dominates u everywhere (using (T GCC))

45 Conclusion and open problems Further result: controllability of the heat equation/exponential lower bound for traces of eigenfunctions Carleman estimates à la Lebeau-Robbiano Some open questions: Fine analysis near the glancing set! Continuity of the solution v L 2 loc(r + ; L 2 (M))? (non-)genericity of the (T GCC) among surfaces satisfying (GCC)? Geometric condition for having both: φ Σ L 2 (Σ) Ce cλ φ L 2 (M), and νφ Σ L 2 (Σ) Ce cλ φ L 2 (M)? Geometric condition for having both: φ Σ L 2 (Σ) C φ L 2 (M), and λ 1 νφ Σ L 2 (Σ) C φ L 2 (M)? for λ λ 0.

46 Thank you!