Control from an Interior Hypersurface

Control from an Interior Hypersurface Matthieu Léautaud École Polytechnique Joint with Jeffrey Galkowski Murramarang, microlocal analysis on the beach March, 23. 2018

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

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.

First Break

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]

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)!

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]

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).

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.

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)

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).

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]

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).

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)).

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 12-13- 17] [Dyatlov-Zworski 12], Torus [Hezari-Rivière 18]

Second break

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?

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)

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}.

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!

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.

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!

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?

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)

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)

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)

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

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)

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

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.

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.

Thank you!