ON THE WELL-POSEDNESS AND REGULARITY OF THE WAVE EQUATION WITH VARIABLE COEFFICIENTS

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1 ESAIM: COCV Vol. 13, N o 4, 007, pp DOI: /cocv: ESAIM: Control, Optimisation and Calculus of Variations ON THE WELL-POSEDNESS AND REGULARITY OF THE WAVE EQUATION WITH VARIABLE COEFFICIENTS Bao-Zhu Guo 1, and Zhi-Xiong Zhang 1, 3 Abstract. An open-loop system of a multidimensional wave equation with variable coefficients, partial boundary Dirichlet control and collocated observation is considered. It is shown that the system is well-posed in the sense of D. Salamon and regular in the sense of G. Weiss. The Riemannian geometry method is used in the proof of regularity and the feedthrough operator is explicitly computed. Mathematics Subject Classification. 35J50, 93C0, 93C5. Received January 1, 006. Revised May 5, 006. Published online September 5, Introduction and main results In the last two decades, the class of well-posed and regular linear systems has been studied extensively [5,3]. It has been demonstrated that this class of systems is quite general: it covers many control systems described by partial differential equations with actuators and sensors supported at isolated points, sub-domains, or on a part of the boundary of the spatial region. More importantly, this class of infinite-dimensional systems, although the input and output operators are allowed to be unbounded, may possess many properties that are parallel in many ways to those of finite-dimensional systems. In particular, the concept of regularity is a completely new concept in this framework. However, while the abstract theory has been quite fruitful, these properties are rarely checked for control systems described by partial differential equations [4]. The well-posedness and regularity of a multidimensional heat equation with both Dirichlet and Neumann type boundary controls were established in [3]. For a wave equation with constant coefficients, boundary Dirichlet input and collocated output on a bounded open domain of R n with smooth boundary, the well-posedness was proved in [1] and the regularity was proved recently in [11]. The regularity of multidimensional Schrödinger and Euler-Bernoulli equations with certain types of control and observation were shown in [9] and [10], respectively. Other results on the well-posedness and regularity of control systems described by multidimensional partial differential equations can be found in [0, ], etc. Most of the aforementioned multidimensional partial differential equations are with constant coefficients. In this paper, we generalize the results of [1, 11] to the variable coefficients case. The system is described by the Keywords and phrases. Wave equation, transfer function, well-posed and regular system, boundary control and observation. Supported by the National Natural Science Foundation of China and the National Research Foundation of South Africa. 1 Academy of Mathematics and Systems Science, Academia Sinica, Beijing , P.R. China; bzguo@iss.ac.cn School of Computational and Applied Mathematics, University of the Witwatersrand, Wits 050, Johannesburg, South Africa. 3 Graduate University of Chinese Academy of Sciences, Beijing , P.R. China. c EDP Sciences, SMAI 007 Article published by EDP Sciences and available at or

2 WELL-POSEDNESS AND REGULARITY OF NON-UNIFORM WAVE EQUATION 777 following wave equation with variable coefficients, partial boundary Dirichlet control and collocated observation: wx, t w tt x, t a ij x =0,x=x 1,x,...,x n,t>0, x i x j wx, t =0,x 1,t>0, 1.1 wx, t =ux, t, x 0,t>0, yx, t = A 1 w t x, t,x 0,t>0, where R n n is an open bounded region with smooth boundary =:= , 1 are disjoint parts of the boundary relatively open in withint 0. Awx, t := and for some constant a>0, a ij x =a ji x C R n, n ν A := ν k a k1 x, k=1 k=1 x i wx, t a ij x,da =H H0 1 x j a ij xξ i ξ j a ξ, x, ξ=ξ 1,ξ,...,ξ n C n, 1. ν k a k x,..., ν k a kn x, k=1 := ν i a ij x x j, 1.3 where ν =ν 1,ν,...,ν n is the unit normal of pointing towards the exterior of. u is the input function or control and y is the output function or observation. Let H = L H 1 and U = L 0. The following theorem is the generalization of Proposition. of [1] for the system 1.1 with constant coefficients. Theorem 1.1. Let T>0, w 0,w 1 Hand u L 0,T; U. Then there exists a unique solution w, w t C[0,T]; H to the system 1.1, which satisfies w, 0 = w 0 and w t, 0 = w 1. Moreover, there exists a constant C T > 0, independent of w 0,w 1,u, such that w,t,w t,t H + y L 0,T ;U C T [ ] w 0,w 1 H + u L 0,T ;U. Theorem 1.1 implies that the open-loop system 1.1 is well-posed in the sense of D. Salamon with the state space H and with the input and output space U [11]. From this result and Theorem. of [] see also Th. 3 of [8], we know that the system 1.1 is exactly controllable in some time interval [0,T] if and only if its closed-loop system under the output proportional feedback u = ky, k > 0 is exponentially stable. We can thereby say that the conditions for the exponential stabilization [15] and for the exact controllability [19] of the wave equation with constant coefficients should be the same. Moreover, for the case of variable coefficients, the condition for the exact controllability stated in Theorem 1.1 of [4] is also the condition for the exponential stability of the closed-loop system 1.1 under the feedback u = ky, k > 0. This fact leads to the following interesting example that comes from [4] and was studied numerically in [7]. Example 1.1. Let n =and a 11 x 1,x =a x 1,x =1+x 1 + x,a 1 x 1,x =a 1 x 1,x =0.

3 778 B.-Z. GUO AND Z.-X. ZHANG Suppose B 1, where B 1 is the unit ball of R. From Example 4.1 of [4] we know that in this case, the system 1.1 is not exactly controllable even if the control is imposed on the whole boundary: 0 =. By the equivalence between exact controllability and exponential stability just explained above, we can now say that for this example, there exists an initial datum w, 0,w t, 0 Hfor which the closed-loop system 1.1 under the output proportional feedback u = ky is not exponentially stable for any k > 0. Theorem 1.. The system 1.1 is regular in the sense of Weiss [1]. Moreprecisely,ifw, 0 = w t, 0 = 0 and u,t u U is a step input, then the corresponding output y satisfies lim σ σ σ 0 yx, tdt ν A x g ux dx =0, 1.4 where ν A x g = g ij x ν k a ki x ν l a lj x, g ij x = Ax 1,Ax =a ij x, x. k=1 l=1 Theorems 1.1 and 1. ensure that the system 1.1 is a well-posed regular linear system with feedthrough operator Dux = ν A x g ux for any u U. The main contributions of this paper are: a generalizing the results of [1, 11] for the system 1.1 where the coefficients are constant to the variable ones; b much simplifying the regularity proof of [11] by using multiplier method on Riemannian manifolds. The remaining part of the paper are organized as follows. In Section, we cast the system 1.1 into an abstract setting studied in [] and [8]. The proofs of Theorems 1.1 and 1. will be presented in Sections 3 and 4, respectively.. Collocated formulation of the system 1.1 Let H = H 1 be the dual space of the Sobolev space H0 1 with usual inner product. Let A be the positive self-adjoint operator in H induced by the bilinear form a, defined by Af, g H 1 H 1 0 = af,g = a ij x f g dx, f,g H0 1 x j x i where a ij x are given by 1.. By means of the Lax-Milgram theorem, A is a canonical isomorphism from DA =H 1 0 onto H. It is easy to show that Af = Af whenever f H H 1 0 and that A 1 g = A 1 g for any g L. Hence A is an extension of A to the space H 1 0. ItcanbeeasilyshownthatDA 1/ =L and A 1/ is an isomorphism from L onto H. Define the map Υ LL 0,H 1/ [15] by Υu = v if and only if { Avx =0,x, vx =0,x 1 ; vx =ux, x 0..1 By virtue of the above map, one can write 1.1 as ẅ + Aw Υu =0.. Since DA isdenseinh, soisda 1/. We identify H with its dual H. Then the following relations hold: DA 1/ H = H DA 1/.

4 WELL-POSEDNESS AND REGULARITY OF NON-UNIFORM WAVE EQUATION 779 An extension à LDA1/, DA 1/ ofa is defined by So. can be further written in DA 1/ as Ãf, g DA 1/ DA 1/ = A 1/ f,a 1/ g H, f,g DA 1/..3 ẅ + Ãw + Bu =0, where B LU, DA 1/ isgivenby Define B LDA 1/,Uby Bu = ÃΥu, u U..4 B f,u U = f,bu DA 1/ DA 1/, f DA1/,u U. Then for any f DA 1/ andu C 0 0, we have f,bu DA 1/ DA 1/ = f,ãã 1 Bu DA 1/ DA 1/ = A1/ f,a 1/ à 1 Bu H = A 1/ f,a 1/ Υu H = f,υu L A = AA 1 1 f f,υu L =,u U..5 In the last step, we have used the fact a ij v x j φ x i dx =0, φ H 1 0, which holds for any classical solution v of.1. Since C 0 0isdenseinU = L 0, we finally obtain that B = A Now, we have formulated the open-loop system 1.1 into an abstract form of a second-order system in the state space H = L H 1 : { ẅt+ Ãwt+But =0, yt = B.7 ẇt, where B and B are defined by.4 and.6, respectively. The abstract system.7 has been studied in detail in [] and [8]. 3. Proof of theorem 1.1 In this section, we generalize Proposition. of [1] to the variable coefficients case. While most of places of the proof are the same to Theorem 1.1 of [16] for the constant coefficients case, we would rather give the proof in detail for the sake of completeness. In what follows, the C T is used frequently to denote some positive constant that is independent of y, u although it may change values from different contexts. We rewrite 1.1 with zero initial data by the operator A

5 780 B.-Z. GUO AND Z.-X. ZHANG into the form: w tt + Aw =0in 0,, w, 0 = 0, w t, 0 = 0 in, w = u on 0 0,, 3.1 y = A 1 w t on 0 0,. By Propositions 3. and 3.3 of [] see also [3], Theorem 1.1 is equivalent to saying that the solution to 3.1 satisfies y L 0,T ;U C T u L 0,T ;U, u L 0,T; U. 3. By considering u 1 = 0, we may assume without loss of generality that 0 ==. Let z := A 1 w t,where w is a solution of 3.1. Since y = B w t = A 1 w t = z, 3. is equivalent to or where z satisfies B w t L 0,T ;L C T u L 0,T ;L 3.3 z C T u L 0,T ;L, 3.4 L 0,T ;L z tt + Az =Υu t in 0,, z, 0 = 0, z t, 0 = 0 in, z =0on 0,. Let Σ = 0,T,Q= 0,T. And denote L Σ := H 0 Σ = L 0,T; L, H 1 Σ := L 0,T; H 1 H 1 0,T; L, H 1 Σ := H 1 Σ. The following lemma is Theorem.3 of [17] in the variable coefficients case, which has already been indicated in the same paper. Lemma 3.1. Assume that F L 1 0,T; H 1, Φ 0 L, Φ 1 H 1, u L Σ and à is a second-order uniformly strongly elliptic operator: where for some α>0 ã ij x =ã ji x C R n, νã := n ν k ã k1 x, k=1 k=1 à := x i ã ij x, x j ã ij xξ i ξ j α ξ, x, ξ=ξ 1,ξ,...,ξ n C n, ν k ã k x,..., ν k ã kn x. k=1 3.5

6 WELL-POSEDNESS AND REGULARITY OF NON-UNIFORM WAVE EQUATION 781 Then the solution Φ to the following equation Φ tt + ÃΦ =F in Q, Φ, 0 = Φ 0, Φ t, 0 = Φ 1 in, Φ=u on Σ 3.6 satisfies Φ, Φ t C[0,T]; L H 1, Remark 3.1. In Lemma 3.1, one can show that Φ, Φ t and Similar remarks apply to all subsequent regularity results. Φ νã Φ H 1 Σ. νã depend continuously on the given datum. Apply Lemma 3.1 to 3.1 to get the regularity: w, w t C[0,T]; L H 1. This produces in turn the regularities of z, the solution of 3.5: { z C[0,T]; H 1 0, Az= w t C[0,T]; H 1, z t = A 1 w tt = A 1 [ Ãw + ÃΥu] = w +Υu L 0,T; L 3.7. Like Lemma 3.1, the following lemma is a direct corollary of Theorem. of [17] in the variable coefficients case. Lemma 3.. For the problem 3.6, if then Φ νã H 1 Σ. F L 1 0,T; H 1 0, Φ0 =Φ 1 =0,u=0, Lemma 3.3. Let Ψ beafunctiondefinedonq. SupposeΨ=0on Σ. Then for s = 1 or s =0, Ψ νã H s Σ Ψ ν Hs Σ. Moreover, there exists a constant C>0 such that C 1 Ψ ν Ψ Hs Σ νã Proof. Since Ψ = 0 on Σ, it has Ψ x j assumption, we may assume that = ν j Ψ ν H s Σ C Ψ ν Hs Σ Ψ on Σ for any 1 j n, andso νã C 1 sup x ã ij ν i ν j C = n ãijν Ψ i ν j ν on Σ. By for some C>0. We only show that Ψ H 1 Σ = Ψ νã ν H 1 Σ and Ψ ν H 1 Σ C Ψ νã H 1 Σ since the proofs for other cases are similar. Now, for any h H0 1 Σ, it has Ψ Ψ h = n ãijν dσ Ψ h i ν j Σ ν hdσ Σ νã C Ψ νã H 1 Σ h H 1 0 Σ. νã H 1 Σ n ãijν i ν j H 1 0 Σ This yields Ψ ν H 1 Σ C Ψ νã H 1 Σ. The proof is complete.

7 78 B.-Z. GUO AND Z.-X. ZHANG Proof of Theorem 1.1. The proof will be split into four steps. Step 1. Let u L Σ. Then by Lemma 3.1, the solution to 3.1 satisfies w, w t C[0,T]; L H 1 w, H 1 Σ. 3.8 Σ Since Υu L 0,T; H 1/, it follows that [13] Substituting 3.8 and 3.9 into 3.7 yields Υu L 0,T; H 1 H 1 Σ. 3.9 z t = w + Υu H 1 Σ. Since z =0onΣ,sodoesz t on Σ. Therefore, it follows from Lemma 3.3 that z t ν H 1 Σ Step. Like the case of constant coefficients in [11], we can use the transform of geodesic normal coordinates to change locally and to :={x, y R n,x > 0,y R n 1 } and :={x, y R n,x =0,y R n 1 }. Under such a change of coordinates, the operator A is changed locally to  := D x + rx, ydy lot [18] or Cor. C.5.3 of [1], where lot denotes the first-order differential operators and rx, ydy stands for the second-order tangential strongly elliptic operator in y variable. Denote by ŵ for w, the solution of 3.1, and by û for u under this change of coordinates. Since ŵ has zero initial data, one can also extend ŵt to be zero for t<0. Let φ C0 R, φ 1, be a smooth cutoff function in R with φt =0fort 3/T and φt =1 while t [0,T] and put v := ŵφ. Then v satisfies v tt = Âv = A 0v + lotŵ in 0,, v, 0 = v t, 0 = 0 in, v = φû on 0,, suppv [0, 3/T ], where A 0 := Dx + rx, ydy is the principal part of Â. Denote Σ := 0,T. Now, decompose v = ϕ + ψ, whereϕ, ψ satisfy 3.11 and 3.13 below, respectively. By Lemma 3.1, the solution ϕ to the following equation ϕ tt + A 0 ϕ =0in 0,, ϕ, 0 = ϕ t, 0 = 0 in, ϕ = φû on 0,, 3.11 satisfies ψ satisfies the following equation ϕ, ϕ t C[0,T]; L H ψ tt + A 0 ψ = f in 0,, ψ, 0 = ψ t, 0 = 0 in, ψ =0on 0,, 3.13

8 WELL-POSEDNESS AND REGULARITY OF NON-UNIFORM WAVE EQUATION 783 where f = lotŵ. Recalling that w C[0,T]; L by 3.8 and so ŵ C[0,T]; L, we obtain, by Lemma 3.1, that This together with 3.1 gives f C[0,T]; H 1 and hence ψ, ψ t C[0,T]; L H v, v t C[0,T]; L H 1. Step 3. We show that for the nonhomogeneous problem 3.13, the map û B ψ t : is continuous from L Σ to L Σ Indeed, since the map û f = lotŵ is continuous from L Σ to L 0,T; H 1, it suffices to show that f B ψ t : is continuous from L 0,T; H 1 to L Σ Apply A 1 0 to 3.13 to give where Ψ := A 1 0 ψ satisfies, by 3.14, that Ψ tt + A 0 Ψ=A 1 0 f in 0,, Ψ, 0 = Ψ t, 0 = 0 in, Ψ=0on 0,, 3.17 Ψ H H 1 0, A 1 0 f L 0,T; H 1 0, A 1 0 ψ t C[0,T]; H 1 0. Apply Lemma 3. to problem 3.17 to obtain Ψ 0 H 1 Σ and so Ψ t 0 L Σ. Finally, by.6, B ψ t = B A 0 A 1 0 ψ t = B A 0 Ψ t = Ψt 0, it follows from Remark 3.1 that A 1 Ψt 0 f 0 = B ψ t is continuous from L 0,T; H0 1 to L Σ then follows from a trivial fact that f A 1 0 f is continuous from L 0,T; H 1 to L 0,T; H0 1. Step 4. The proof will be accomplished if we can show that for problem 3.11, the map û B ϕ t : is continuous from L Σ to L Σ Comparing problem 3.1 with problem 3.11 and noticing the equivalences of 3., 3.3 and 3.4, we find that 3.18 is equivalent to ẑ C T û L Σ, where ẑ := A 1 0 ϕ t L Σ Let X x, y, t S 0 R be a pseudodifferential operator with smooth symbol of localization χx, y, t, η, σ supported in the elliptic sector of := Dt D x rx, yd y, where the principal part of the D Alambertian is written in local coordinates. Here the dual variables η R n 1,σ R correspond to the Fourier s variables of y iη, t iσ. Suppose suppχ {x, y, t, η, σ R R n 1 R, σ C 1 η }, and supp1 χ {x, y, t, η, σ R R n 1 R, σ C η }, where0<c <C 1 are constants. We show that I X ẑ ν L Σ. 3.0

9 784 B.-Z. GUO AND Z.-X. ZHANG The trick comes from the application of Actually, by 3.10, it has 1+σ + η 1 iσ z ν L R n η,σ, where z is the partial Fourier transform of ẑ respect to y, t. Hence Rn σ z 1+σ + η ν dη dσ < +. Since when σ C η, η 1, it has C C + C 1 η + C +1 = C η 1+ η C + η σ 1+σ + η, hence z +C σ σ C η ν dη dσ z σ C η C 1+σ + η ν dη dσ +C Rn σ z C 1+σ + η ν dη dσ <. Therefore 1 χ z R ν dη dσ = 1 χ z n supp1 χ ν dη dσ z σ C η ν dη dσ <. Thus 3.0 is valid. Now, we need to show that X ẑ ν L Σ. This is a little bit easy due to its ellipticity. Returning to problem 3.11 for ϕ, rewritten as ϕ = 0 and applying X, we will see that the variable X ϕ satisfies X ϕ = [X, ]ϕ H 1 Q, X ϕ Q L Q, 3.1 where and henceforth we take for Q an extend cylinder based on [ T,T ] and denote Σ := [ T,T ]. Indeed, by the fact that [X, ] S 1 Q and the priori regularity in 3.1, we have [X, ]ϕ H 1 Q. Furthermore, X ϕ Σ = X φû L Σ. And by the pseudolocal property of pseudodifferential operators and the fact that suppϕ [0, 3/T ], we have X ϕt, C and X ϕ T, C. This yields the boundary condition X ϕ Q L Q in 3.1. Since X is a pseudodifferential elliptic operator, apply the classical elliptic theory to the elliptic problem 3.1 to yield X ϕ H 1/ Q+H 1 Q =H 1/ Q, 3. where the first term in the middle of 3. is due to the boundary regularity of 3.1, and the second term is due to the interior regularity. Next, we return to the elliptic problem { Az = wt in Q, z Σ =0

10 WELL-POSEDNESS AND REGULARITY OF NON-UNIFORM WAVE EQUATION 785 from 3.7. Since ẑ := A 1 0 ϕ t in Q, the counterpart of the above elliptic problem in the half-space Q is locally { A0 ẑ = ϕ t in Q, Apply X to above problem to yield ẑ Σ =0. A 0 X ẑ = X ϕ t +[A 0, X ]ẑ = d dt X ϕ [ d dt, X ] ϕ +[A 0, X ]ẑ. Notice that [A 0, X ] S 1 Q and[ d dt, X ] S0 Q. By the priori regularity in 3.1 for ϕ and in 3.7 for z, we conclude that [ ] d dt, X ϕ +[A 0, X ]ẑ L Q. 3.3 Moreover, by virtue of 3., X ϕ H 1, 1 Q H 0, 1 Q,wherewehaveusedtheanisotropicHörmander s spaces of [1] on page 477. In the space H m,s Q, m is the order in the normal direction to the plane x =0 which plays a distinguished role and m + s is the order in the tangential direction in t and y. Since d dt is d a first order differential operator in the tangential direction, dt X ϕ H 0, 1 Q H 1,0 Q =H 1 Q. By 3.3, we are led to solving the following problem { A0 X ẑ H 1/ Q+L Q H 1/ Q, X ẑ Σ =0. Since A 0 X is elliptic in Q, by the classical elliptic regularity again, we obtain Combining 3.4 and 3.0 yields X ẑ H 3/ Q, ẑ ν Finally, since ẑ =0on Σ, it follows from Lemma 3.3 that ν X ẑ L Σ. 3.4 =I X ẑ ν + X ẑ ν L Σ. ẑ 0 L Σ, proving The proof is complete. 4. Proof of Theorem 1. Although here we consider the variable coefficients case, the proof below is much simpler than the one in [11] for the case of constant coefficients. This is due to the use of multiplier method on Riemannian manifolds, which was an effective tool introduced in [4]. In the case of constant coefficients, the multipliers are reduced to the usual ones in Euclidean spaces. Notice the hypothesis 1. and set Ax :=a ij x, Gx :=g ij x = Ax 1, Gx :=detg ij x.

11 786 B.-Z. GUO AND Z.-X. ZHANG Let R n be the usual Euclidean space. For each x =x 1,x,...,x n R n, define the inner product and norm over the tangent space R n x = R n by gx, Y := X, Y g = g ij xα i β j, X g := X, X 1/ g, X = i=1 α i,y= x i i=1 β i R n x x. i It is easily checked that R n,g is a Riemannian manifold with Riemannian metric g. Denote by D the Levi- Civita connection with respect to g. Let H be a vector field on R n,g. Then for each x R n,thecovariant differential DH of H determines a bilinear form on R n x Rn x : DHX, Y = D X H, Y g, X, Y R n x, where D X H stands for the covariant derivative of vector field H with respect to X. For any f C R n,x = n i=1 α i x i,denote g f := g f := f x i a ij x x j, div g X := i=1 1 Gxaijx f Gx x i x j 1 Gx x i Gxα i x, = Af + T f, T := Now, it follows from the Appendix of [8] that the transfer function of the system.7 is a ij x Gx Gx x i x j Hλ =λb λ + Ã 1 B, 4.1 where Ã, B and B are given by.3,.4 and.6, respectively. Moreover, from the well-posedness claimed by Theorem 1.1, it follows that there are constants M,β > 0 such that [6] sup Hλ LU = M<. 4. Reλ β Proposition 4.1. Theorem 1. is valid if for any u C0 0, the solution w to the following equation λ wx = a ij x w x = Awx, x, x i x j wx =0,x 1, wx =ux, x satisfies lim 1 wx λ R,λ + ν A g ux 0 λ dx =0. Proof. It was shown in [1] that in the frequency domain, 1.4 is equivalent to lim λ R,λ + Hλu = ν A g u in the strong topology of U for any u U, 4.4

12 WELL-POSEDNESS AND REGULARITY OF NON-UNIFORM WAVE EQUATION 787 where Hλ is given by 4.1. Due to 4. and a density argument, it suffices to show that 4.4 is satisfied for all u C0 0. Now assume that u C0 0, and put Then w satisfies 4.3 and wx = λ + Ã 1 Bux. Hλux = λ A 1 w x, x Take a function v H to satisfy Then 4.3 can be written as λ wx x i a ij x vx =0,x, x j vx =0,x 1 ; vx =ux, x 0. w v =0, x i a ij x wx vx =0,x, x j or λ A 1 wx =wx vx. So 4.5 becomes Hλux = 1 wx 1 vx 4.6 λ λ Since vx is independent of λ, the required result then follows from 4.6 and 4.4. The following lemma, which may be useful for other purposes, is the generalization of Lemma.1 in [14], p. 18, which refers to Euclidean spaces, to the Riemannian manifold,g. Lemma 4.1. Suppose is of class C k,k 1. Assume 1. and 1.3. Then there exists a vector field N : R n of class C k 1 such that where µ := metric g. Nx =µx, x ; N g 1, x 4.7 νa ν A g is the unit normal of pointing towards the exterior of in terms of the Riemannian Proof. Since isofclassc k,k 1, for every fixed x 0, there is an open neighborhood V of x 0 in R n and a function φ : V R of class C k such that Replacing φ by φ if needed, we may assume that g φx 0, x V and φx =0iffx V. µx 0, g φx 0 g > 0.

13 788 B.-Z. GUO AND Z.-X. ZHANG Then the function ψ := g φ/ g φ g : V R n is of class C k 1. We show that ψ = µ on V. In fact, since φ =0onV, we have φ φ = ν j on V, j=1,,...,n. x j ν Hence g φ = φ a ij = x i x j φ ν i ν a ij = φ x j ν ν i a ij = φ x j ν ν A, which implies that ψ, g φ, ν A and µ are parallel to each other on V. This together with ψ g = µ g =1 shows that ψ = µ on V. Since is bounded, iscompactinr n ; therefore can be covered with a finite number of neighborhoods V 1,V,...,V m. Each of them plays the role of V in the earlier reasoning. Denoting by ψ i,i =1,,...,m the corresponding functions of V i,wehave V 1 V... V m and ψ i = µ on V i, i=1,,...,m. Fix an open set V 0 R n such that V 0 V 1 V m,v 0 = and define ψ 0 : V 0 R n by ψ 0 x = 0 for all x V 0. Let θ 0,θ 1,...,θ m be a partition of unity of class C k, corresponding to the covering V 0,V 1,...,V m of : θ i C k 0 V i and0 θ i 1,i=0, 1,...,m and It is obvious that is the required vector field. θ 0 + θ θ m =1on. m N := θ i ψ i i=0 Proof of Theorem 1.. We define ũ C as the extension of u, i.e. ũ = u on 0 and ũ =0on 1.Set F λũ := 1 w λ µ, x, where w satisfies 4.3 with u C0 0. µ = νa ν A g and µ is defined similarly as in 1.3: n i=1 µ i x i = 1 ν A g. The proof will be accomplished if we can show that µ := lim F λũ =ũ on λ R,λ L, 4.8 which will be split into two steps. Step 1. Let the vector field N be given by 4.7. Since ũ C, the solution w to 4.3 belongs to C and hence Nw C. Multiply the both sides of the first equation of 4.3 by Nw and integrate by parts. By formulae for divergence on the Riemannian manifold,g: div g w N=N w + w div g N, 4.9

14 WELL-POSEDNESS AND REGULARITY OF NON-UNIFORM WAVE EQUATION 789 and we obtain LHS = Re λ wnwdx div g w Ndx = w N,µ g d, = λ = λ = λ By the Green s formula on the Riemannian manifold,g: g wnwdx = g w, g Nw g dx N w dx = λ [div g w N w div g N]dx w N,µ g d λ w div g Ndx ũ d λ w div g Ndx. Nw w µ d and Lemma.1 of [4], we have RHS = AwNwdx = g wnwdx T wnwdx = g w, g Nw g dx + Nw w µ d T wnwdx = g w, g Nw g dx + w µ d T wnwdx = DN g w, g wdx 1 g w g N,µ gd + 1 g w g div gndx + w µ d T wnwdx = DN g w, g wdx 1 g w gd + w µ d + 1 g w gdiv g Ndx T wnwdx = 1 w µ T ũ g d + [ g w div g N g DN g w, g w]dx T wnwdx, wherewehaveusedthefactnw = w µ on, and T is the gradient of the tangential on : g w g = w µ + T ũ g on. In what follows, we use the same C to denote the constant that is independent of λ and w although it may have different values in different contexts.

15 790 B.-Z. GUO AND Z.-X. ZHANG Letting LHS=RHS gives the identity: F λũ d = ũ d + 1 λ T ũ gd + 1 fλ, 4.10 λ where fλ =Re DN g w, g wdx λ w + g w g div gndx +Re T wnwdx satisfying Indeed, fλ fλ Cλ w L + gw g L DN g w, g w dx + λ w + g w g div gn dx + T w Nw dx D gwn, g w g dx + Cλ w L + gw g L + T w dx + Nw dx D g w g w g dx + Cλ w L + gw g L N g w g g + C g w g L + gw g L C g w g L + Cλ w L + gw g L +C gw g L Cλ w L + gw g L, wherewehaveusedthefactssup div g N C, T w C g w g and Nw g w g when N g 1. x Next, multiply the right side of the first equation of 4.3 by w and integrate by parts to have This together with 4.3 gives Aw w dx = = g w w dx T w w dx w g w, g w g dx + µ ũd T w w dx. 1 w Re F λũ, ũ L =Re λ µ ũd = 1 g w, g w g dx λ 1 Aw w dx + 1 λ λ Re T w w dx = 1 g w g dx + λ w dx + 1 λ λ Re T w w dx.

16 WELL-POSEDNESS AND REGULARITY OF NON-UNIFORM WAVE EQUATION 791 On the other hand, by the Cauchy-Schwartz inequality, one has Re T w w dx C λ w dx + 1 λ Hence Re F λũ, ũ L Combining 4.11 and 4.1, we obtain g w g dx. 1 C λ w L λ + 1 λ gw g L. 4.1 fλ λ C λ λ w L + 1 λ gw g L C λ C Re F λũ, ũ L C λ C F λũ L + ũ L Finally, from inequality 4.13 and identity 4.10, we conclude that Step. Putting lim λ R,λ F λũ L = ũ L we deduce from 4.14 that Gλũ := Re F λũ, ũ L, λ > 0, 4.15 lim sup λ Next, the formula 4.1 implies that Gλũ lim sup F λũ L ũ L = ũ L. λ lim λ w L = lim λ w L Re F λũ, ũ L lim ũ L lim λ λ λ λ C λ λ C =0. On the other hand, integrate the both sides of 4.9 over to yield ũ L =Re wnwdx + w div g Ndx λ w L + 1 λ gw g L + C w L λ λ C Gλũ + C w L, where we have used 4.1 and the fact Nw g w g when N g 1. This together with the fact lim λ w L =0gives ũ L lim inf λ λ Gλũ = lim inf λ C Gλũ. λ We have thus obtained that lim λ ũ L then follows from 4.14 and The proof is complete. Acknowledgements. The authors would like to thank Professor P. F. Yao for his kind help in the proof of Theorem 1. and analogous referees for their helpful comments.

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