Carleman estimates for the Euler Bernoulli plate operator

Transcription

1 Electronic Journal of Differential Equations, Vol. 000(000), No. 53, pp ISSN: URL: or ftp ejde.math.swt.edu ftp ejde.math.unt.edu (login: ftp) Carleman estimates for the Euler Bernoulli plate operator Paolo Albano Abstract We present some a priori estimates of Carleman type for the Euler Bernoulli plate operator. As an application, we consider a problem of boundary observability for the Euler Bernoulli plate coupled with the heat equation. 1 Introduction In [1] Tataru the author proved some estimates with singular weights for the heat for the wave equations. These estimates are a powerful tool for the study of observability ( controllability) of pde s. The present paper is concerned with a problem of boundary observability for the following system w tt + w=α θ in ]0,T[ Ω θ t θ =β w in ]0,T[ Ω w = ν w = w=θ=0 on [0,T] Ω (1) here α, β R, Ω is an open domain in R n with smooth boundary Ω, T is a positive number by ν =(ν 0,...,ν n ) we denote the unit outer normal vector to [0,T] Ω. We are concerned with the following question. Knowing that ν w ν θ areequalto0on[0,t] Ω can we conclude that w θ are 0 in [0,T] Ω? More precisely, for any solution (w, θ) of problem (1), we want to prove an observability estimate of the form (w, θ) C ( ν w, ν θ) where are suitable interior boundary Sobolev norms. In other words, this would say that the solution (w, θ) can be reconstructed in a stable fashion if one observes some derivatives on the boundary. Note that the initial data cannot be recovered stably due to the parabolic regularizing effect. Mathematics Subject Classifications: 93B07, 74F05. Key words: Carleman estimates, observability, thermo elasticity. c 000 Southwest Texas State University University of North Texas. Submitted March 14, 000. Published July 1,

2 Carleman estimates EJDE 000/53 However, the final data at time T can be obtained. By duality this implies an exact null controllability result for the adjoint problem. The affirmative answer to the previous questions is given using a priori estimates of Carleman type for the heat equation for the following problem w tt + w=f in ]0,T[ Ω w = ν w = w=0 on [0,T] Ω. () Our goal is to derive an estimate of the form e τφ w C( ν w + e τφ f ) for solutions of (). The Carleman estimates have been introduced in [3], were intensively studied in [5] (for hyperbolic elliptic operators) in [7] (in the case of anisotropic operators). A general approach to these estimates for boundary value problems was developed in [11, 13]. Carleman estimates for the initial boundary value problem for the heat equations have been independently obtained in [4, 6] [14] while the analogous results in the case of hyperbolic equations were proved in [1]. We remark that other approaches to the observability problem for hyperbolic equations have been developed in [8] (using multipliers method) in [] (using microlocal analysis). For what concerns observability estimates for the heat equations, an observability result was proved in [9] using microlocal analysis Carleman estimates for elliptic equations. A similar result was proved in [1] in the case of the wave operator instead of the plate operator. The additional difficulty here is due to the higher order of the operators we deal with. We also remark that since both the heat the plate operators are anisotropic (the weights of the time space derivatives are 1 respectively), no lower bound on the observation time, T, is required. Finally, for simplicity we will limit our computations to the constant coefficients case, but similar results hold true also in the case of operators of the same type with C 1 principal parts L lower order terms. We start recalling a Carleman estimate with singular weight for the heat equation. Next, we deduce a similar estimate for the plate operator (decoupling such an operator as the product of two Schrödinger ones). Finally, putting together the previous estimates we get an observability estimate for the coupled system, which, in particular, yields an affirmative answer to our question. We point out that the last step will require a precise control on the constant that appear on the Carleman estimates. Notation Preliminaries Denote by (t, x) (or(x 0,x 1,...,x n )) the coordinates in [0,T] Ω. We call t the time variable, while the other n coordinates are called the space variables. By, R, we denote the L scalar product its real part respectively.

3 EJDE 000/53 Paolo Albano 3 We will use the following shorter notation D j = 1 i x j, u i 1...i k = i1... ik u. The symbol u t indicates the derivative of u with respect to t, represents the gradient with respect to the space variables while D = i 1 is its selfadjoint version u is the Hessian matrix of u (w.r.t. the space variables). Given two operators, A B, as usual we define their commutator as [A, B] :=AB BA. By H s we denote the classical Sobolev spaces, with norm s while sts for the L norm. In the Carleman estimates we use weighted Sobolev norms. Set α =(α 0,α 1,...,α n )=(α 0,α ). Given a nonnegative function η define the following anisotropic norm u = η (k α a) α u dt dx k,η α a k where α = α αn n α a =α 0 +α α n. We conclude this section recalling a Carleman estimate for the heat equation. Consider the parabolic initial boundary value problem θ t (t, x) θ(t, x) =f(t, x) in ]0,T[ Ω, θ =0 on [0,T] Ω, θ(0,x)=θ 0 (x) in Ω, (3) with (f,θ 0 ) L ([0,T] Ω) H0 1 (Ω). A Carleman estimate for the solution of the above problem looks like e τφ θ e τφ ν θ + e τφ f, with appropriate norms a suitable function φ, uniformly with respect to the large parameter τ. The obstruction to such an estimate is that no Sobolev norm of the initial data can be controlled by the right h side. Hence the only hope is to consider a weight function φ which approaches at time 0. Estimates of this type have been proved in ([1, 4, 6] [14]). Thus, we introduce a function g defined as 1 g(t) = (t ]0,T[). (4) t(t t) Notice that ( d ) kg(t) Ck g k (t) t ]0,T[ (5) dt for a suitable positive constant C k. Let ψ(x) be a function such that ψ(x) 0 forallx Ω 1. (6) 1 This is a pseudoconvexity condition with respect to the heat operator.

4 4 Carleman estimates EJDE 000/53 Define φ(t, x) :=g(t) ( e λψ(x) e λφ ), Φ= ψ L (Ω). (7) The weight function φ thus defined approaches at times 0,T.The additional parameter λ is essential in order to obtain the control of the constants which enables us to hle arbitrarily large coefficients in the coupling terms. Theorem.1 Let φ be given as in (7) with ψ satisfying (6). Letθ be the solution of problem (3). Then there exists λ 0 so that for each λ>λ 0 there exists τ λ so that for τ>τ λ the following estimate holds uniformly in λ, τ: C 1 λ τ 1 e τφ θ eτφ f + τe τφ, τ ν ψ ν θ dσ, (8) here τ = λτge λψ C 1 is a suitable positive constant. For the reader convenience we give the proof of the above result in the appendix, for a proof in a more general setting see [1]. 3 The Plate Operator Let us consider the following problem w tt + w=f in ]0,T[ Ω w = ν w = w=0 on [0,T] Ω (w(0,x),w t (0,x)) (H 4 (Ω) H0 (Ω)) H (Ω), (9) with f L ([0,T] Ω). Our goal is to prove a Carleman estimate for a solution w of the above problem. The idea of the proof is the following. First, we decompose the plate operator as the product of two Schrödinger ones. Then, we get some Carleman estimates for such operators (see Theorem 3.1 Theorem 3.3 below). Finally, the result follows putting together these estimates (see Theorem 3.4). It is clear that in order to prove a Carleman estimate for the Schrödinger operator we need to assume a suitable condition on the weight function. More precisely, we suppose a positive constant γ exists so that, for any x Ω, ψ(x)ξ ξ γ ξ ξ R n. (10) Lemma 3.1 Let u C(]0,T[; H (Ω)) C 1 (]0,T[; L (Ω)) be a solution of the (ill posed) problem u t + i u = f in ]0,T[ Ω (11) u=0 on [0,T] Ω. This is a strong pseudoconvexity condition for the constant coefficients Schrödinger equation.

5 EJDE 000/53 Paolo Albano 5 with f L (]0,T[ Ω) let φ be given as in (7) with ψ satisfying (6) (10). Then there exists λ 0 so that for each λ>λ 0 there exists τ λ so that for τ>τ λ the following estimate holds uniformly in λ, τ: C e τφ τ 1 u 1, τ τe τφ ν ψ ν u dσ + e τφ f, (1) here τ = λτge λψ C is a suitable positive constant. Proof. Let us compute the conjugated operator to P = t +i. We have that P τ = e τφ Pe τφ = id t τφ t id + iτ φ + τ(d φ+ φ D). Clearly, we can split the operator P τ into its symmetric antisymmetric parts as follows P τ = P s τ + P a τ with P s τ = τφ t +τ(d φ+ φ D) Pτ a = id t id + iτ φ. Set v = e τφ u observe that v(0,x)=v(t,x)=0 x Ω (13) v(t, x) =0 (t, x) [0,T] Ω. (14) So, estimate (1) reduces to C τ 1 v 1, τ τ ν ψ ν v dσ + e τφ f. (15) Clearly, P τ v = P s τ v + P a τ v +R P s τ v, P a τ v (16) R P s τ v, P a τ v =τr ( φ t + D φ+ φ D)v, i(d t D + τ φ )v. In order to complete the proof we must only to compute explicitly the terms of the above scalar product. It is immediate to see that Integrating by parts using (14), we get R φ t v, iτ φ v =0. (17) τr (D φ+ φ D)v, iτ φ v = τ 3 v, φ φ v (18) = 4τ 3 φ φ φv,v.

6 6 Carleman estimates EJDE 000/53 Moreover, recalling (13), we have that τr φ t v, id t v = τr [ φ t,id t ]v, v = τ φ tt v, v (= lower order term), (19) while (13) (14) imply that τr (D φ+ φ D)v, id t v = τr( φ Dv, id t v + (D D t φ+d φd t )v, iv ) (0) = τr( φ Dv, id t v + D D t φv, iv + D t v, i φ Dv ) = τr φ t Dv, v (= lower order term). (Here in the sequel, lower order terms means terms which can be absorbed in the LHS of (15) taking τ large enough.) Finally, integrating by parts once recalling (14), we get τr φ t v, id v = τr φ t Dv, v (= lower order term). (1) Now, it remains to compute the higher order terms. Using once more integrations by parts (14) we obtain that τr (D φ+ φ D)v, id v = τr D k ((D φ+ φ D)v),iD k v 4τ = τr τ = τr k=1 (D (D k φ)+ (D k φ) D)v),iD k v k=1 ν φ ν v dσ ν φ ν v dσ () (D φ k + φ k D)v),D k v τ k=1 ( = τr τ φdv Dv + 1 i v ( φ) Dv dt dx ) ν φ ν v dσ Substituting (17) () in (16), we get P τ v 4τ 3 φ φ φv,v +τ φ tt v, v 4τR φ t Dv, v +τ R (D j φ)d j v, v τ ν φ ν v dσ +4τ j=1 ν φ ν v dσ φ jk D k v, D j v j,k=1 R (4τ 3 φ φ φ+τφ tt (4τ φ t ) (τ ( φ) ) )v, v + 4τ φdv Dv Dv dt dx τ ν φ ν v dσ.

7 EJDE 000/53 Paolo Albano 7 Moreover, it is not difficult to verify that 4τ 3 φ φ φ+τφ tt (4τ φ t ) (τ ( φ) ) λ τ 3 4τ φ I τi provided that λ τ are sufficiently large. So, P τ v + τ ν ψ ν v dσ C τ 1 v, 1, τ provided that λ τ are sufficiently large. Theorem 3. Let u, φ ψ be as in Lemma 3.1. Then there exists λ 0 so that for each λ>λ 0 there exists τ λ so that for τ>τ λ the following estimate holds uniformly in λ, τ: C 3 e τφ u 1, τ e τφ ν ψ ν u dσ + e τφ τ 1 f, (3) here τ = λτge τψ C 3 is a suitable positive constant. Proof. Set P = t + i. Applying Lemma 3.1 to τ 1 u we obtain C e τφ u 1, τ e τφ ν ψ ν u dσ + e τφ ([P, τ 1 ]+ τ 1 P)u, for large enough τ the commutator is small compared to the LHS therefore we get C 3 e τφ u 1, τ e τφ ν ψ ν u dσ + e τφ τ 1 f, the conclusion follows. The next result follows arguing as in Lemma 3.1 in Theorem 3.. Theorem 3.3 Let u C(]0,T[; H 4 (Ω)) C 1 (]0,T[; H (Ω)) C (]0,T[ L (Ω)) be a solution of the (ill posed) problem u t i u = f in ]0,T[ Ω u= ν u=0 on [0,T] Ω (4) let φ be given as in (7) with ψ satisfying (6) (10). Then there exists λ 0 so that for each λ>λ 0 there exists τ λ so that for τ>τ λ the following estimate holds uniformly in λ, τ: C 4 e τφ u 1, τ eτφ τ 1 f, (5) here τ = λτge τψ C 4 is a suitable positive constant.

8 8 Carleman estimates EJDE 000/53 Now, the main result of this section can be easily proved. Theorem 3.4 Let w be the solution of the problem (9) let φ be given as in (7) with ψ satisfying (6) (10). Then there exists λ 0 so that for each λ>λ 0 there exists τ λ > 0 so that for τ>τ λ the following estimate holds uniformly in λ, τ: C e τφ w, τ e τφ ν ψ ν w dσ + e τφ τ 1 f, (6) here τ = λτge λψ C is a suitable positive constant. Proof Clearly, problem (9) can be recast as follows w t i w = u in ]0,T[ Ω w = ν w =0 on [0,T] Ω u t + i u = f in ]0,T[ Ω u=0 on [0,T] Ω. Now, applying Theorem 3.3 to the solution of problem (7) we get C 4 e τφ w 1, τ e τφ τ 1 u whilst using Theorem 3., we deduce that the solution of (8) satisfies C 3 e τφ u 1, τ e τφ ν ψ ν w dσ + e τφ τ 1 f (7) (8) (9) (30) for sufficiently large λ τ. Now, the conclusion follows putting together (9) (30). Remark 3.5 We note that a stronger estimate could be proved. In fact, the interior (microlocal) Carleman estimate for the plate operator shows that one can replace the H (anisotropic) norm in the LHS of (6) with the H 3 (anisotropic) norm (see e.g. [1]). 4 The observability estimate In this section we prove the observability result for the system (1). Theorem 4.1 Suppose that (w, θ) is a solution of the system (1) let φ be given as in (7) with ψ satisfying (6) (10). Then there exists λ 0 so that for each λ>λ 0 there exists τ λ > 0 so that for τ>τ λ the following estimate holds uniformly in λ, τ ( e τφ w + eτφ τ 1, τ θ C e τφ, τ ν ψ ν w dσ ) + τe τφ ν ψ ν θ dσ, (31) for a suitable positive constant C.

9 EJDE 000/53 Paolo Albano 9 Remark 4. This estimate shows that solutions to (1) can be reconstructed in a stable manner if one observes their normal derivative on the boundary. As one can see from the above estimate, this observation needs not be on the entire boundary; it suffices to observe (w, θ) in the region Γ, given by Γ=[0,T] {x Ω; ν ψ(x) > 0}. Proof of Theorem 4.1: Applying Theorem 3.4 to w we deduce that C e τφ w e τφ, τ ν ψ ν w dσ + e τφ τ 1 α θ. (3) On the other h, Theorem.1 implies that λc 1 e τφ τ 1 θ eτφ β w + τe τφ, τ ν ψ ν θ dσ. (33) Then, the conclusion follows by adding (3) (33) provided that λ is sufficiently large. Combining the straightforward energy estimates 3 with (31) we obtain the following consequence: Theorem 4.3 For all solutions (w, θ) of the system (1) we have θ(t ) L (Ω) + w t(t ) L (Ω) + w(t ) L (Ω) C( ν w L (Γ) + νθ L (Γ) ) for a suitable positive constant C. The next example shows in the case of Ω a ball of R n how one can choose the function φ. Example 4.4 Let Ω = {x R n : x < 1} fix x R n \ Ω. Then, define ψ(x) = x x 1 φ(t, x) = t(t t) (eλψ(x) e λ sup y Ω ψ(y) ). It is immediate to verify that φ satisfies all the assumption of Theorem 4.1. Moreover, if (w, θ) is a solution of equation (1) ν w ν θ are equal to 0on Γ={(t, x) [0,T] Ω:x (x x)>0}, then w θ are equal to 0 in ]0,T[ Ω. 3 Here the energy is defined by E(t) = 1 Ω θ(t, x) + w t (t, x)+ w(t, x) dx.

10 10 Carleman estimates EJDE 000/53 A Proof of Theorem.1 Set Q(D) = t. As usual, with the substitution v = e τφ θ, the estimate (8) reduces to C 1 λ τ 1 v Q, τ τv + τ ν ψ ν v dσ, (34) where Q τ is the conjugated operator defined as Notice that Split Q τ into where Q τ (t, x, D) :=e τφ Q(D)e τφ. v(0,x)=v(t,x)=0 v(t, x) =0 x Ω (t, x) [0,T] Ω. Q τ (t, x, D) =Q a τ (t, x, D)+Qs τ (t, x, D) Q a τ (t, x, D) = t+τ( φ + φ ) is the skew-symmetric part of Q τ while is the symmetric part of Q τ. Since Q s τ (t, x, D)v = τ φ τφ t Q τ v = Q s τ v + Q a τ v + Q s τ v, Qa τv, (35) the crucial step of the proof will be to estimate from below Q s τ v, Qa τ v = ( + τ φ + τφ t )v, ( t + τ( φ+ φ ))v integrating by parts. It is easy to see that t v, Q s τv is a lower order term compared to the left h side in (34) since therefore t v, Q s τ v = [Qs τ, t]v, v = + ( t (τ φ + τφ t )v, v t v, Q s τv c λ τ g 3/ v, for a suitable positive constant c λ. Similarly, for some c λ > 0, τ( φ+ φ )v, τφ t v c λ τ g 3/ v.

11 EJDE 000/53 Paolo Albano 11 Then it remains to estimate τ ( φ+ φ )v, ( + τ φ )v. Compute first the leading zero order terms. Integrating by parts, we get τ ( φ+ φ )v,τ φ v =τ 3 v, φ φ )v. Since φ = λge λψ ψ, the highest order terms occur when the derivative falls on the exponential. Hence we get τ ( φ+ φ )v, τ φ v =4λ τ 3 ψ 4 v, v +R (36) where R sts for lower order terms, R = O( τ 1/ v, τ ). Now compute the leading first order terms integrating by parts the expression τ ( φ+ φ )v, v. Since one operator is symmetric the other is skew-symmetric, we need to compute their commutator. In fact, we have that τ v, ( φ+ φ )v = τ ( φ+ φ ) v, v τ v, ( φ+ φ )v. Hence, using the equality τ v, ( φ+ φ )v = τ +τ we deduce that Then, τ v, ( φ+ φ )v = τ [, ( φ+ φ )]v, v τ τ [, ( φ+ φ )]v, v =τ = τ = 4τ φ v v+ v (( φ+ φ )v)dt dx ν φ ν v dσ ν φ ν v dσ. ( φ k + φ k )v k,v k=1 v k ( φ k v)dt dx k=1 φ v vdtdx+τ ( φ)v, v

12 1 Carleman estimates EJDE 000/53 τ v, ( φ+ φ )v = 4 τλ ψ v dt dx + R τ ν ψ ν v dσ. (37) Hence, if we put together (36) (37) then we get 4λ τ 3 v + τ ψ v dt dx Q s τ v, Q a τv + τ ν ψ ν θ dσ + R. Substituting this in (35) we obtain 4λ τ 3 v + τ ψ v dt dx + Q s τ v + Q a τ v Q τ v + τ ν ψ ν θ dσ + R. In the first term on the left we already control the appropriate weighted L norm of v. The corresponding norm of t v is easily obtained from the second the fourth term, while the weighted L norms of v, respectively v are obtained in an elliptic fashion from the first the third term to obtain Cλ τ 1 v Q, τ τv + τ ν ψ ν θ dσ + R. Now the lower order terms in R are negligible (i.e. much smaller than the left h side) for sufficiently large λ, τ we obtain (34). References [1] P. Albano, D. Tataru Carleman estimates boundary observability for a coupled parabolic-hyperbolic system, Electronic J. of Differential Equations 000 (000), no., 1-15 (electronic). [] C. Bardos, G. Lebeau, J. Rauch Controle et stabilisation de l equation des ondes, SIAM J. Control Optim. 30 (199), no. 5, [3] T. Carleman Sur un problém pour les systèmes d équations aux derivées partielles à deux variables independantes, Ark. Mat. Astr. Fys. B, 6 (1939), 1-9. [4] A. V. Fursikov, O. Yu. Imanuvilov Controllability of evolution equations, Lecture Notes Series, 34, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996.

13 EJDE 000/53 Paolo Albano 13 [5] L. Hörmer The Analysis of Linear Partial Differential Operators, Springer Verlag, [6] O. Yu. Imanuvilov Controllability of parabolic equations, Sbornik Mathematics, 186 (1995), no. 6, [7] V. Isakov Carleman type Estimates in Anisotropic Case Applications, Journal of Differential Equations, 105 (1993), no., [8] J.L. Lions Controlabilité exacte, stabilisation et perturbations the systèmes distribuès, Tome 1 Tome,Masson, [9] G. Lebeau, L. Robbiano Exact control of the heat equation, Comm. Partial Differential Equations, 0 (1995), no. 1-, [10] G. Lebeau, E. Zuazua Null Controllability of a System of Linear Thermoelasticity, Arch. Rational Anal., 141 (1998), no. 4, [11] D. Tataru A priori estimates of Carleman s type in domains with boundary, Journal de Mathematiques Pure et Appl., 73 (1994), [1] D. Tataru Boundary controllability for conservative P.D.E, Applied Math. Optimization, 31 (1995), [13] D. Tataru Carleman estimates unique continuation near the boundary for P.D.E. s, Journal de Math. Pure et Appl., 75 (1996), [14] D. Tataru Carleman Estimates, Unique Continuation Controllability for anizotropic PDE s, Contemporary Mathematics, Volume 09 (1997), Paolo Albano Dipartimento di Matematica, Università di Roma Tor Vergata Via della Ricerca Scientifica, Roma, Italy albano@axp.mat.uniroma.it