UNIQUE CONTINUATION ESTIMATES FOR THE LAPLACIAN AND THE HEAT EQUATION ON NON-COMPACT MANIFOLDS

Transcription

1 UNIQUE CONTINUATION ESTIMATES FOR THE LAPLACIAN AND THE HEAT EQUATION ON NON-COMPACT MANIFOLDS LUC MILLER Abstract This article concerns some quantitative versions of unique continuation known as observability inequalities One of them is a lower bound on the spectral projectors of the Dirichlet Laplacian which generalizes the unique continuation of an eigenfunction from any open set Ω Another one is equivalent to the interior null-controllability in time T of the heat equation with Dirichlet condition the input function is a source in, T ) Ω) On a compact Riemannian manifolds, these inequalities are known to hold for arbitrary T and Ω This article states and links these observability inequalities on a complete non-compact Riemannian manifold, and tackles the quite open problem of finding which Ω and T ensure their validity It proves that it is sufficient for Ω to be the exterior of a compact set for arbitrary T ), but also illustrates that this is not necessary It provides a necessary condition saying that there is no sequence of balls going infinitely far away from Ω without shrinking in a generalized sense depending on T ) which also applies when the distance to Ω is bounded Contents 1 Introduction 1 11 Controllability/observability for the heat equation 2 12 Observability at low-frequencies for the heat and the Laplacian 2 13 Carleman inequalities and interpolation 3 14 The theorems 3 2 Sufficient condition: i) ii) 4 3 From interpolation to observability: ii) iii) 6 4 From the Laplace to the heat equation: iii) iv) 7 5 From low-frequencies to full observability: iv) v) 7 6 Necessary condition: v) vi) 8 7 Finer sufficient condition: i ) ii) 8 References 9 1 Introduction Let M be a smooth connected complete n-dimensional Riemannian manifold with metric g and boundary M When M, M denotes the interior and M = M M Let denote the negative) Dirichlet Laplacian on L 2 M) with domain D ) = H 1 M) H 2 M) Let Ω be an open subset of M and let T be a positive time 2 Mathematics Subject Classification 35B6, 35B37, 58J35, 58J5, 93B7 Submitted to the Mathematical Research Letters on April 29, 24 1

2 2 L MILLER To begin with, we formulate various quantitative versions of unique continuation on M for the heat equation and the spectral projectors 1 µ, with references to their introduction and proof on a compact M Besides their intrinsic significance, they have applications to the controllability of distributed systems and the estimation of its cost eg heat conduction, plate vibrations) Then we state the main result which addresses the quite open problem of their validity in this non-compact setting by describing their logical relationship and by providing some sufficient and some necessary geometric conditions 11 Controllability/observability for the heat equation Consider the heat equation on M with Dirichlet boundary condition and a source term located in Ω T =, T ) Ω 1Ω T denotes its characteristics function): 1) t φ φ = 1Ω T f on R t M, φ) = φ L 2 M), f L 2 locr; L 2 M)) φ = on R t M, Definition 1 The heat equation on M is said to be null-controllable in time T from Ω if for all φ L 2 M) there is a control function f L 2 R M) such that the solution φ C [, ), L 2 M)) of 1) satisfies u = at t = T By duality cf [DR77]), it is equivalent to the following quantitative version of unique continuation for solutions ut, x) = e t u x) of the heat equation without source term: t u = u on R t M, u = on R t M Definition 2 The heat equation on M is said to be final-observable in time T from Ω if there is a positive constant C T,Ω such that: 2) u L 2 M), e T u L 2 M) C T,Ω e t u L 2 Ω T ) The best constant C T,Ω in 2) can be considered as the controllability cost since it is also the best constant such that f L 2 Ω T ) C T,Ω u L 2 M), for all intial condition u and control f solving the controllability problem in def1 When M is compact, Lebeau and Robbiano proved in [LR95] that these controllability/observability properties hold for arbitrary T and Ω) In [MZ3], it is mentioned that their approach does not apply to unbounded Euclidean domains indeed Weyl s asymptotics for eigenvalues and Russell s construction of biorthogonal functions are used in [LR95]) and that the only result available on these specific properties is that they hold when M is a domain of the Euclidean space with the flat metric and the exterior of Ω is bounded cf [CDMZ1]) To this open problem, this article contributes the adaptation of the Lebeau-Robbiano approach to a non-compact M, the extension of the sufficient condition in [CDMZ1] and the necessary condition in [Mil4b] to a non-euclidean M, and a finer sufficient condition for a homogeneous M which applies even if the exterior of Ω is not compact 12 Observability at low-frequencies for the heat and the Laplacian In [LR95], Lebeau and Robbiano take advantage of the fast damping of high frequencies to reduce def2 to the following estimate on the cost of fast null-controllability at low-frequencies Definition 3 Fast observability at low-frequencies for the heat equation on M from Ω holds if there are positive constants D 1 and D 2 such that for all τ, 1] and µ 1: 3) u 1 µ L2 M), e τ u L2 M) D 2 τ e D1µ e t u L2 Ω τ )

3 UNIQUE CONTINUATION ESTIMATES ON NON-COMPACT MANIFOLDS 3 When M is compact, this improvement of prop1 in [LR95] allowing γ to take the endpoint value γ = 1) is a corollary of prop2 in [LZ98] In [LR95], the parabolic problem was reduced to a space-time elliptic one following a classical idea in the uniqueness for parabolic equations cf [Lin9]) The improvement in [LZ98] was obtained by first reducing def3 to the following lower bound for 1Ω 1 µ, ie the restriction of the spectral projectors to Ω Definition 4 Observability at low-frequencies for the Laplacian on M from Ω holds if there are positive constants C 1 and C 2 such that for all µ > : 4) v 1 µ L2 M), v L 2 M) C 2 e C1µ v L 2 Ω) When M is compact, this is an inequality on sums of eigenfunctions proved as th3 in [LZ98] and th146 in [JL99] In particular, when v is an eigenfunction of, this is a quantitative version of elliptic unique continuation This spectral property should have wide applications besides the study of nodal sets cf [JL99]) and the control of vibrations cf [Mil4a]) 13 Carleman inequalities and interpolation In [LZ98] and [JL99], this observability at low-frequencies for the Laplacian is reduced to the following interpolation inequality for the space-time Laplacian P = + 2 t on M T =, T ) M with Dirichlet condition on the parabolic boundary Γ = R t M) {} M) and with observation of the Neumann condition on {} Ω Definition 5 The Laplacian P satisfies the interpolation inequalities in time T from Ω if α, 1/2), δ, 1), C >, ψ H 2 M T ) st ψ = on Γ, ψ H 1 αt,1 α)t ) M) C ψ δ 1 δ 5) H 1 M T ) P ψ L 2 M T ) + t ψ t= L Ω)) 2 When M is compact, this is lemmaa in [LZ98] and lemma145 in [JL99] it is also related to lemma43 in [Lin91]) Following classical ideas in unique continuation for elliptic equations, the interpolation inequalities are based on Carleman-type estimates The needed type, namely Carleman estimate for boundary problems, were introduced by Robbiano in [Rob95] The specific Carleman inequalities used in [LZ98] and [JL99] are proved in prop1 and prop2 of [LR95] 14 The theorems Before stating the main result, we introduce a geometric condition The fundamental tone λ B) of an open subset B of M is: λ B) = ϕ C ϕ 2 inf L2 M) B)\{} ϕ 2 L 2 M) When B is smooth and compact eg B is a ball with radius lower than the injectivity radius), then λ B) is just the first eigenvalue of the Dirichlet Laplacian on B in the classical sense In particular, the Euclidean ball B r with radius r satisfies λ B r ) = λ B 1 )/r 2, and the balls Bx, r) of center x M and radiusxs r satisfy λ Bx, r)) λ B 1 )/r 2 as r Cf eg sect39 in [Cha93]) Definition 6 The sequence of B k ) k N of open subsets of M goes away from Ω without shrinking with a factor T if dist T B k, Ω) 2 4T 2 λ B k ) + as k +, where dist T is the following averaged distance with Gaussian weight of variance T : ) dist T B, Ω) 2 distx, B)2 = 2T log exp dx distb, Ω) 2 2T volω) 2T Ω\B In [Mil4b], a necessary condition similar to def6 was proved in the Euclidean case using pointwise rather than L 2 Gaussian upper bounds) and illustrated by elementary examples In particular, it applies even if the distance to Ω is bounded

4 4 L MILLER Remark 1 Assuming that λ κ,r is the fundamental tone of the sphere of radius r in the space form of constant sectional curvature κ R and that the Ricci curvatures of M are all greater than or equal to n 1)κ, we have λ B) λ κ,r for every ball B of radius r cf [Che75] or th324 of [Cha93]) If the volume of Ω is finite and the radii of the balls B k are all greater than a positive constant, then def6 just says that the geodesic distance from B k to Ω tends to infinity Theorem 1 Each of the following statements implies the next one: i) Ω is the non-empty exterior of a compact set K su ch that K Ω M = ii) 2 t + satisfies the interpolation inequalities in time T from Ω def5) iii) Observability at low-frequencies for the Laplacian on M from Ω holds def4) iv) Fast observability at low-frequencies for the heat on M from Ω holds def3) v) The heat equation on M is final-observable in time T from Ω def2) vi) There is no sequence B k ) k N of open subsets of M going away from Ω without shrinking with a factor T def6) To obtain a sufficient condition for these observability inequalities that would be further from the case of a compact M than i) in th1, one is lead to make some assumptions on the geometry of M For instance, if M is a homogeneous manifold, a uniform version of def5 holds: for any given R > r > and α, 1/2), δ, 1), C >, x M, ψ H 2 M T ) st ψ = on Γ, 6) ψ H 1 αt,1 α)t ) B R ) C ψ δ H 1 M T ) P ψ L 2 M T ) + t ψ t= L 2 B r)) 1 δ, where B r and B R are the balls with center x and respective radii r and R Indeed, for fixed x this interpolation inequality results from lem23, and δ and C do not depend on x since M is homogeneous The next theorem relaxes the assumption that the uncontrolled region M \ Ω be compact Theorem 2 If the Ricci curvature of M is bounded below and the uniform interpolation inequalities 6) hold eg if M is homogeneous), then th1 holds with i) replaced by: i ) There are positive constants r and d such that any point in M is at a distance smaller than d of some ball of radius r included in Ω It would be interesting to know which boundedness assumptions on the geometry of M ensure 6) 2 Sufficient condition: i) ii) The key to the proof of i) ii) is the following global Carleman inequality with boundary term, similar to the local one in prop1 of [LR95] Taking advantage of the structure of P, it provides unique continuation with respect to t only, but in terms of integral quantitities on the whole manifold M Let F be the space of functions f C R M) such that suppf), T ) M and f [,T ] M = Let F be the space of functions f F such that f t= = Proposition 21 Let ϕ : R R be a smooth function of time such that ϕt) and ϕt) > for all t [, T ] There are positive constants h and C such that for all h, h ) and g F : h 2 P g 2 2ϕ e h + h h t g 2 7) e 2ϕ h Ch g 2 + h t g 2 + h g 2) e 2ϕ h M T {} M Proof For each f F, let f = f t= Let f, g) = M T fḡ, f 2 = f, f), f, g) = M f ḡ, f 2 = f, f) We also introduce standard semiclassical notations where h, 1) is a small) parameter: D t = h i t, x = h i, x = h 2, f 2 1 = M T

5 UNIQUE CONTINUATION ESTIMATES ON NON-COMPACT MANIFOLDS 5 f 2 + D t f 2 + x f 2 With f = e ϕ/h g and P ϕ = h 2 e ϕ/h P e ϕ/h, the Carleman inequality 7) writes: 8) P ϕ 2 + h D t f 2 Ch f 2 1 We denote by ϕ k the derivative of order k of ϕ, and let ϕ k = inf [,T ] ϕ k, ϕ k = sup [,T ] ϕ k As usual, we introduce the hermitian and anti-hermitian parts of P ϕ : A = P ϕ + Pϕ = Dt 2 + x ϕ 2 1 and B = P ϕ Pϕ = D t ϕ 1 + ϕ 1 D t = 2ϕ 1 D t hϕ 2, 2 2i Since P ϕ = A + ib, we have P ϕ f 2 = Af 2 + Bf 2 + i[bf, Af) Af, Bf)] Since by integration by parts g, Af) = Ag, f) ihg, D t f) and g, Bf) = Bg, f) for all f F and g F, we deduce that: 9) P ϕ f 2 = Af 2 + Bf 2 + i[a, B]f, f) + 2hϕ 1 D t f, D t f) The computation of the commutator of A and B, after the simplifications: [A, B] = [D 2 t ϕ 2 1, D t ϕ 1 + ϕ 1 D t ] = D t [D 2 t, ϕ 1 ] + [D 2 t, ϕ 1 ]D t + ϕ 1 [D t, ϕ 2 1] + [D t, ϕ 2 1]ϕ 1, yields i[a, B] = 4hϕ 2 D 2 t + ϕ 2 1) h 3 ϕ 4 = 4hϕ 2 A x + ϕ 2 1) h 3 ϕ 4 Therefore, i[a, B]f, f) = 4h Af, ϕ 2 f) + ϕ 2 x f ϕ 1 f 2) h 3 ϕ 4 f, f) Since Af, 4hϕ 2 f) Af 2 /4 + 4h) 2 ϕ 2 f 2, we deduce: 1) i[a, B]f, f) h 4ϕ 2 x f 2 + 8ϕ 2 16h 1 ϕ2 2) f 2) Af 2 /4 Since 2 ϕ 1 D t f, 2hϕ 2 f) ϕ 1 D t f 2 + 2h) 2 ϕ 2 f 2, we have: Bf 2 = 2ϕ 1 D t f 2 + hϕ 2 f 2 2 Reϕ 1 D t f, 2hϕ 2 f) ϕ 2 1 D tf 2 5h 2 ϕ 2 2 f 2 Plugging this inequality and 1) in 9) yields: P ϕ f 2 3/4)) Af 2 + hc h f hϕ 1 D t f, D t f) hc h f 2 1 2h ϕ 1 D t f 2, { } with C h = min 8ϕ 2 21h 1 ϕ2 2, 4ϕ 2, ϕ 2 Taking h 1 > small enough, so that C h > yields 8) for all h, h ), which completes the proof of prop21 This global Carleman inequality implies as in lemma3 of [LR95]) the following global interpolation inequality whith the notations of def5: Lemma 22 Let U be an open subset of M such that U Ω τ, T ), δ, 1), C >, ψ H 2 M T ) st ψ = on Γ, ψ H 1 U τ ) C ψ δ H 1 Ω T \Ω τ ) P ψ L 2 M T ) + ψ H 1 Ω\U) τ ) + t ψ t= L 2 Ω)) 1 δ Proof Let τ τ, T ) and let χ 1 : R [, 1] be a smooth function such that χ 1 t) = 1 for t τ and χ 1 t) = for t τ Let χ 2 : M [, 1] be smooth function such that suppχ 2 ) Ω and χ 2 = 1 on U Applying prop21 with φt) = exp t) and g = χψ where χ = χ 1 χ 2, using h < 1 and dividing by h 3, yields positive constants C and h < 1 such that for any ψ as in lem22 and all h, h ): h M T P ψ, [P, χ]ψ) 2 e 2ϕ h + {} Ω t ψ 2 e 2ϕ h C ψ, t ψ, ψ) 2 2ϕ e U T Since ϕ is decreasing, φ) = 1, supp[p, χ]) Ω \ U) τ Ω τ \ Ω τ ) and [P, χ] is a differential operator of order one, there is a C 1 > depending on χ such that: he P ψ 2 L 2 M T ) + hc 1e ψ 2 H 1 Ω\U) τ ) + hc 1e 2ϕτ ) h ψ 2 H 1 Ω τ \Ω τ ) + e tψ t= 2 L 2 Ω) C e 2ϕτ) h ψ 2 H 1 U τ ), for all h, h ), h

6 6 L MILLER Therefore, there are constants h 1, h ), D >, D 1 > 1 ϕτ) >, and D 2 = ϕτ) φτ ) > such that: e D 1 ) h P ψ L2 M T ) + ψ H1 Ω\U) τ ) + t ψ t= L2 Ω) + e D 2 h ψ H1 Ω τ \Ω τ ) D ψ H 1 U τ ), for all h, h 1 ) Optimizing with respect to h as in [Rob95] completes the proof of lem22 Gluing together local interpolation inequalities as in sect3b of [LR95]) on a fixed compact set intersecting Ω yields the following global interpolation inequality The first term of its left-hand-side corresponds to 2) in sect3 of [LR95] with t and x interverted, whereas the second term corresponds to 1)) Lemma 23 For all compact subsets K and K such that K M, for all segment S, T ), τ, T ), δ, 1), C >, ψ H 2 M T ) st ψ = on Γ, ψ H 1 K τ ) + ψ H 1 S K ) C ψ δ H 1 M T ) P ψ L 2 M T ) + t ψ t= L 2 Ω)) 1 δ Assume i), ie M \ K = Ω for some compact subset K of M such that K Ω M Let V be an open subset of M containing K Ω such that V is compact and V M Applying lem22 to U = Ω \ V using Ω T \ Ω τ M T and Ω \ U V, and applying lem23 to K = V, K = K, S = [αt, 1 α)t ] and τ = 1 α)t yield: δ, 1), C >, ψ H 2 M T ) st ψ = on Γ, 11) 12) 13) ψ H 1 Ω 1 α)t ) C A δ B + ψ H 1 V 1 α)t ) ψ H1 V 1 α)t ) C A δ B 1 δ, ψ H 1 αt,1 α)t ) K) C A δ B 1 δ, ) 1 δ with A = ψ H1 M T ) and B = P ψ L2 M T ) + t ψ t= L2 Ω) Plugging 12) into 11) as in lem4 of [LR95] yields the estimate ψ H1 Ω 1 α)t ) C A δ B 1 δ Since αt, 1 α)t ) M Ω 1 α)t αt, 1 α)t ) K, adding the square of this estimate to the square of 13) yields def5 with δ = max {δ, δ } and C large enough Remark 24 Even when K Ω M, the statement of def5 avoids the corner {} M, hence it should still be true and tractable When K M, choosing a V containing K avoids using 13) and yields an estimate of ψ on M 1 α)t rather the smaller set αt, 1 α)t ) M in def5 3 From interpolation to observability: ii) iii) Let µ >, ϕ CcompM) and v = 1 µ ϕ Since C compm) is dense in L 2 M), it is enough to prove 4) for such v Let de λ denote the projection valued measure associated to the self-adjoint operator The cardinal hyperbolic sine function is the smooth function shc C R) defined by: shc) = and shct) = expt) exp t))/2t) for t Let F µ t, λ) = t shctλ) 1 λ µ For all j and k in N, λ j t k F µ CR t ; L R λ )) Therefore the function ψ defined by: ψt, x) = F µ t, )ϕ = k N t 2k+1 2k + 1)! )k v, satisfies ψ H j X T ) for all j N and T > Moreover, since ϕ D ) j ) for all j N, we have ψt, ) j N D ) j ) H 1 M) for all t We assume that P = t 2 + satisfies the interpolation inequalities in time T from Ω for some α, 1/2), δ, 1), C > Since F µ, λ) = and ψt, ) H 1 M), we have ψ = on Γ, so that 5) applies to ψ We have P ψ =,

7 UNIQUE CONTINUATION ESTIMATES ON NON-COMPACT MANIFOLDS 7 since t 2 F µ t, λ) = λ) 2 F µ t, λ), and t ψ t= = v since t shct)) t= = shc) = 1, so that 5) writes: 14) ψ H 1 αt,βt ) M) C ψ δ H 1 M T ) v 1 δ L 2 Ω), with β = 1 α The left-hand-side of 14) squared is greater or equal to: βt ψ 2 L 2 αt,βt ) M) = F µ t, λ) 2 de λ ϕ, ϕ) dt = Similarly, ψ 2 H 1 M T ) = T αt βt αt ) 2 dt µ αt µ βt µ αt de λ ϕ, ϕ) = 1 2α)α 2 T 3 v 2 L 2 M) t shctλ) 2 de λ ϕ, ϕ) dt 1 + λ 2 ) t shctλ) 2 de λ ϕ, ϕ) dt T 1 + µ 2 )e 2T µ v 2 L 2 M) Plugging these last two estimates in 14) yields, v 2 L 2 M) 1 + µ 2 1 2α)α 2 T 2 e2t µ v 2δ L 2 M) v 21 δ) L 2 Ω) Therefore, for any C 1 > T/1 δ), there is a C 2 > such that 4) holds 4 From the Laplace to the heat equation: iii) iv) Assume that observability at low-frequencies for the Laplacian on M from Ω holds Let τ, 1], µ 1 and u 1 µ L2 M) For all t [, τ], we may apply 4) to v = e t u since it is in 1 µ L2 M): µ C2e 2 2C1µ e t u 2 L 2 Ω) et u 2 L 2 M) = e 2tλ2 de λ u, u ) First integrating on [, τ] with the new variable s = t/τ, then using τ 1 and finally 1 exp αt)dt = 1 exp α))/α 2α) 1 for α ln 2 yields: C 2 2e 2C1µ e t u 2 L 2 Ω τ ) τ 1 τ µ 1 e 2τsλ2 de λ u, u ) ds e 2sµ2 ds µ de λ u, u ) Therefore, for any D 1 > C 1, there is a D 2 > such that 3) holds τ 4µ 2 u 2 L 2 M) 5 From low-frequencies to full observability: iv) v) By duality cf [DR77]), def3 is equivalent to the following null-controllability at low frequencies cf def1): for all τ, 1] and µ 1, there is a bounded operator Sµ τ : L 2 M) L 2 R M) such that for all φ 1 µ L2 M), the solution φ C [, ), L 2 M)) of 1) with control function f = Sµφ τ satisfies 1 µ φ = at t = τ Moreover, we have the cost estimate: Sτ µ D2 τ e D1µ We introduce a dyadic scale of frequencies µ k = 2 k k N) and a sequence of time intervals τ k = σ δ T/µ δ k where δ, 1) and σ δ = 2 k N 2 kδ ) 1 >, so that the sequence of times defined recursively by T = and T k+1 = T k + 2τ k converges to T The strategy of Lebeau and Robbiano in [LR95] is to steer the initial state φ to through the sequence of states φ k = φt k ) 1 >µ k 1 L 2 M) at frequencies converging to infinity by applying recursively the control function f k = S τ k µ k φ k to φ k during a time τ k and zero control during a time τ k This strategy is succesful if φ k tends to zero and the control function ft) = k 1 t T k τ k f k t)

8 8 L MILLER is in L 2, T ; L 2 M)) Since the cost estimate above implies S τ k µ k D 2 e D1µ k / τ k, it only remains to check that: 15) ε k = φ k and C k = De D1µ k /µ δ k satisfy lim k ε k = and k N C 2 kε 2 k < Since 1 µ k φt k +τ k ) =, we have ε k+1 e τ kµ 2 k φtk +τ k ) The expression of φt k + τ k ) in terms of the source term f k Duhamel s formula) and e t 1 contractivity of the heat semigroup) imply φt k + τ k ) 2ε k + τ k f k ) Therefore ε k+1 2e τ kµ 2 k 1 + τk C k )ε k Since C k+1 /C k = e D1µ k /2 δ/2, we deduce: C k+1 ε k+1 2e τ kµ C k ε k 2 k 1 + D2 e D1µ k) e D 1µ k /2 δ/2 D exp 2D 1 µ k σ δ T µ 2 δ ) k, for some D > Since 2 δ > 1 this implies k N C2 k ε2 k <, which proves 15) and completes the proof of the null-controllability in def1 6 Necessary condition: v) vi) The main ingredient is an L 2 Gaussian upper bound on the heat kernel Following [CGT82] and section 62 in the book [Tay96]), the finite propagation speed of solutions to the wave equation yields: 16) 1U e t 1V LL 2 M)) 1 4πt s ρ exp ) s2 ds exp 4t ) ρ2 4t for any open sets U and V in M such that ρ = distu, V ) = inf x,y) U V distx, y) Assume that the sequence B k ) k N of open subsets of M goes away from Ω without shrinking with a factor T def6) For any k N, let k denote the Dirichlet Laplacian on B k with domain D k ) = H 1 B k ) H 2 B k ) and let k denote its extension to L 2 M) by zero Let ũ k L 2 B k ) be an eigenfunction of k with unit norm associated to its first eigenvalue λ B k ) >, and let u k L 2 M) be the extension of ũ k by zero Since k is an extension of, we have: e T u k L2 M) e T k u k L2 M) = e T kũ k L2 B k ) = exp λ B k )T ) Partitioning Ω into slices U = {x Ω ρ distx, B k ) < ρ + ε} of thickness ε >, applying 16) to each U with V = B k, summing up and letting ε yields: e t u k L 2 Ω T ) ) distx, B)2 T exp dx = T exp dist T B k, Ω) 2 ) 2T 4T Ω\B Plugging the last two inequalities in 2) contradicts the limit in def6 Therefore the heat equation on M is not final-observable in time T from Ω 7 Finer sufficient condition: i ) ii) In this section, we prove that th2 results from the following covering lemma obtained by combining the argument of the proof of lem12 in [Shu92] with Gromov s volume comparision theorem prop41iii in [CGT82] or th31 in [Cha93]): Lemma 71 If the Ricci curvature of M is bounded below there are ρ > and N N such that for any ρ, ρ ) there is a sequence of balls B ρ y j ) j N) with center y j M and radius ρ covering M such that each ball B ρ y j ) intersects no more than N other balls B ρ y k ) k i) Let B ρ y j ) j N) be a covering with multiplicity not greater than N as in this lemma Assuming i ), for each j N, there is an x j Ω such that B r x j ) Ω and,

9 UNIQUE CONTINUATION ESTIMATES ON NON-COMPACT MANIFOLDS 9 disty j, B r x j )) d Setting R = r + d + ρ implies B ρ y j ) B R x j ), so that, fixing α, 1/2), applying 6) with x = x j yields j, ψ H 2 M T ) st ψ = on Γ, ψ H1 αt,1 α)t ) B ρy j)) C ψ δ H 1 M T ) P ψ L2 M T ) + t ψ t= L2 Ω)) 1 δ Summing up over j implies: ψ H1 αt,1 α)t ) M) NC ψ δ H 1 M T ) P ψ L 2 M T ) + t ψ t= L 2 Ω)) 1 δ Therefore def5 holds and the proof of i ) ii) is completed References [CDMZ1] V R Cabanillas, S B De Menezes, and E Zuazua, Null controllability in unbounded domains for the semilinear heat equation with nonlinearities involving gradient terms, J Optim Theory Appl 11 21), no 2, [CGT82] J Cheeger, M Gromov, and M Taylor, Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, J Differential Geom ), no 1, [Cha93] I Chavel, Riemannian geometry: a modern introduction, Cambridge Tracts in Mathematics, vol 18, Cambridge UP, Cambridge, 1993 [Che75] S Y Cheng, Eigenvalue comparision theorems and its geometric applications, Math Zeit ), [DR77] S Dolecki and D L Russell, A general theory of observation and control, SIAM J Control Optimization ), no 2, [JL99] D Jerison and G Lebeau, Nodal sets of sums of eigenfunctions, Harmonic analysis and partial differential equations Chicago, IL, 1996), Univ Chicago Press, Chicago, IL, 1999, pp [Lin9] F-H Lin, A uniqueness theorem for parabolic equations, Comm Pure Appl Math ), [Lin91], Nodal sets for solutions of elliptic and parabolic equations, Comm Pure Appl Math ), [LR95] G Lebeau and L Robbiano, Contrôle exact de l équation de la chaleur, Comm Partial Differential Equations ), no 1-2, [LZ98] G Lebeau and E Zuazua, Null-controllability of a system of linear thermoelasticity, Arch Rational Mech Anal ), no 4, [Mil4a] L Miller, How violent are fast controls for Schrödinger and plates vibrations?, Arch Ration Mech Anal ), no 3, [Mil4b], On the controllability of the heat equation in unbounded domains, to appear in Bull Sci Math, preprint, 24 [MZ3] S Micu and E Zuazua, Null-controllability of the heat equation in unbounded domains, Unsolved Problems in Mathematical Systems and Control Theory VD Blondel and A Megretski, eds), Princeton UP, 23, to appear 24 [Rob95] L Robbiano, Fonction de coût et contrôle des solutions des équations hyperboliques, Asymptotic Anal ), no 2, [Shu92] M A Shubin, Spectral theory of elliptic operators on noncompact manifolds, Astérisque 1992), no 27, 5, 35 18, Méthodes semi-classiques, Vol 1 Nantes, 1991) [Tay96] M E Taylor, Partial differential equations I, Applied Mathematical Sciences, vol 115, Springer-Verlag, New York, 1996, Basic theory Équipe Modal X, EA 3454, Université Paris X, Bât G, 2 Av de la République, 921 Nanterre, France address: LucMiller@u-paris1fr