Some uniqueness results for the determination of forces acting over Germain-Lagrange plates

Transcription

1 Some uniqueness results for the determination of forces acting over Germain-Lagrange plates Three different techniques A. Kawano Escola Politecnica da Universidade de Sao Paulo A. Kawano (Poli-USP) Germain-Lagrange plates May / 49

2 Plan Introduction Presentation of the Germain-Lagrange Operator 1 Introduction Presentation of the Germain-Lagrange Operator Presentation of the inverse problems 2 Inverse Problems Unbounded plate Rectangular Plate Plates with arbitrary shapes A. Kawano (Poli-USP) Germain-Lagrange plates May / 49

3 Basic Equation Introduction Presentation of the Germain-Lagrange Operator The Germain-Lagrange equation models the vibration of a plate. Is is given by 2 u t u = h, where u is the displacement and h is the loading. A. Kawano (Poli-USP) Germain-Lagrange plates May / 49

4 Introduction Presentation of the Germain-Lagrange Operator Fundamental solution We can work out an explicit expression for a fundamental solution φ of the Germain-Lagrange operator. It is given by φ(t, x) = H(t) 2π + 0 sen[r 2 t] r J 0 (r x ) dr. (1) A. Kawano (Poli-USP) Germain-Lagrange plates May / 49

5 Introduction Presentation of the Germain-Lagrange Operator The Germain-Lagrange operator is not hypo-elliptic We have two properties: The operator P = is not hypo-elliptic. t 2 Proof The speed of propagation of signals in a Germain-Lagrange plate is infinite. Proof A. Kawano (Poli-USP) Germain-Lagrange plates May / 49

6 Plan Introduction Presentation of the inverse problems 1 Introduction Presentation of the Germain-Lagrange Operator Presentation of the inverse problems 2 Inverse Problems Unbounded plate Rectangular Plate Plates with arbitrary shapes A. Kawano (Poli-USP) Germain-Lagrange plates May / 49

7 Introduction Inverse problems Statement of the problems presented here Presentation of the inverse problems We consider the problem of proving that loads h of the form h(t, x) = g(t)f (x) in the equation 2 u t u = h = g(t)f (x), can be determined uniquely from data about the displacement of a set of points in the plate. In general, f is a distribution and g a function of time that is class C 1. A. Kawano (Poli-USP) Germain-Lagrange plates May / 49

8 Introduction Presentation of the inverse problems Inverse problems Statement of the problems presented here Given the fact that the speed of propagation is infinite, the data for the inverse problem is taken in arbitrary small intervals of time and space. The problems we are going to present are: 1 Unbounded plate that extends over all R 2. 2 Rectangular plate. 3 Bounded plate with arbitrary shape. A. Kawano (Poli-USP) Germain-Lagrange plates May / 49

9 Plan Inverse Problems Unbounded plate 1 Introduction Presentation of the Germain-Lagrange Operator Presentation of the inverse problems 2 Inverse Problems Unbounded plate Rectangular Plate Plates with arbitrary shapes A. Kawano (Poli-USP) Germain-Lagrange plates May / 49

10 Unbounded Plate Inverse Problems Unbounded plate where 2 u + 2 u = g f, in ]0, + ) R 2, t 2 u(0) = 0, (0) = 0, u t f E (R 2 ) (To be determined); T 0 > 0 such that g C([0, T 0 ]) and g(0) 0. (2) A. Kawano (Poli-USP) Germain-Lagrange plates May / 49

11 Inverse Problems Unbounded plate Data for the identification of f E (R 2 ) Theorem 1 (For the unbounded case) If T 0 > 0 such that g C([0, T 0 ]) and g(0) 0, and f E (R 2 ), then for arbitrary 0 < T < T 0, the knowledge of the set Γ ]0,T [ Ω = { u, ψ ϕ : ψ C c (]0, T [), ϕ C c (Ω)}, where Ω R 2 is arbitrary, is enough to determine uniquely f E (R 2 ). A. Kawano (Poli-USP) Germain-Lagrange plates May / 49

12 Idea of the proof Technique: spherical means Inverse Problems Unbounded plate The steps for the proof are: 1 Factor into two Schrodinger operators 2 Write explicitly the solution in terms of a convolution with a fundamental solution of the equation. It is well known that the only solution w C ([0, + ), S (R 2 )), supported in [0, + ), that satisfies (9) is given by w(t, x) = E t f, where the convolution is performed only in the spatial variable, and for t > 0, E t (x) = 1 x 2 4πı t eı 4t. 3 Aplication of a spherical means result A. Kawano (Poli-USP) Germain-Lagrange plates May / 49

13 Plan Inverse Problems Rectangular Plate 1 Introduction Presentation of the Germain-Lagrange Operator Presentation of the inverse problems 2 Inverse Problems Unbounded plate Rectangular Plate Plates with arbitrary shapes A. Kawano (Poli-USP) Germain-Lagrange plates May / 49

14 Rectangular Plate Inverse Problems Rectangular Plate ρh 2 ũ + t 2 D 2 ũ = g Q, in ]0, + ) (]0, L 1 [ ]0, L 2 [), ũ = ũ = 0, at t = 0, t γ R (ũ( t) ) ( ) = γ 2ũ R ( t) = 0, t 0. ν 2 (3) where g : [0, + ) R is a C 1 function with g(0) 0, ũ stands for the vertical displacement, R is the boundary of the rectangle R =]0, L 1 [ ]0, L 2 [, ν is the normal to R, where it is defined, and γ R : H 1 ( R) H 1 2 ( R) is the trace operator u u R. The constants ρ, h, D > 0 that appear in (3) stand respectively for mass density, plate thickness and flexural rigidity. A. Kawano (Poli-USP) Germain-Lagrange plates May / 49

15 Inverse Problems Rectangular Plate Data for the rectangular plate problem The set { Γ ã T ũ( t) =, ψ : t [0, T } a ], ψ C a, Ω c ( Ω), (4) where Ω R is any arbitrary non-empty line segment parallel to one of the rectangle sides, which we take as being the Ox axis, containing any point (x, y 0 ) R such that sen(y 0 n) 0, n N, is enough for the identification of Q L 2 ( R), provided that T a > L 1L 2 4π The set ρh D. Γ c Tc,[0,L 2 ] = { ũ x ( t, 0, ), ψ } : t [0, T c ], ψ Cc ([0, L 2 ]), (5) where T c > 0 can be arbitrarily small, is enough for the determination of Q L 2 ( R). A. Kawano (Poli-USP) Germain-Lagrange plates May / 49

16 Inverse Problems Rectangular Plate Steps in the proof Technique: almost periodic distributions 1. Representation of the loading 2. Representation of the solution 3. Application of the data 4. The main ingredient: Almost Periodic Distributions 5. Application of almost periodic distributions property A. Kawano (Poli-USP) Germain-Lagrange plates May / 49

17 Plan Inverse Problems Plates with arbitrary shapes 1 Introduction Presentation of the Germain-Lagrange Operator Presentation of the inverse problems 2 Inverse Problems Unbounded plate Rectangular Plate Plates with arbitrary shapes A. Kawano (Poli-USP) Germain-Lagrange plates May / 49

18 Inverse Problems Plates with arbitrary shapes Plates with arbitrary shapes Let Ω R 2 be any bounded subset with smooth and regular boundary. Let T 0 > 0 and N N. Consider the plate equation 2 u t u = N n=1 g n f n, in ]0, T 0 [ Ω, u = u t = 0, at t = 0, γ Ω (u(t)) = γ Ω ( u(t)) = 0, t [0, T 0 [, where the set {g n : n {1,..., N}} C 1 ([0, T 0 ]) is linearly independent, and f n H 1 (Ω), n {1,..., N}. The vertical displacement is represented by u, and γ Ω : H 1 ( R) H 1 2 ( R) is the trace operator u u Ω. (6) A. Kawano (Poli-USP) Germain-Lagrange plates May / 49

19 Inverse Problems Plates with arbitrary shapes Data for plates with arbitrary shapes Let O R 2 be any open neighborhood of G + (x) R 2. Now define VG + (x) = O Ω. Also, define for any x 0 R 2 \ Ω and T < T 0 the set Γ T,G+(x 0 )(u) = {(t, φ, u(t, ), φ ) : t [0, T ], φ C c (VG + (x 0 ))}. (7) Observe that from Γ T,G+(x0 )(u) it is not possible to extract information about derivatives of u with respect to the space variable at the boundary, since all φ Cc (VG + (x 0 )) are already null near Ω. Γ T,G+(x0 )(u) is in fact interior data. Define for any x 0 R 2 \ Ω and T < T 0 the set {( ) } u(t, ) ˆΓ T,G+(x0 )(u) = t, φ,, φ : t [0, T ], φ Cc (G + (x 0 )), ν (8) A. Kawano (Poli-USP) Germain-Lagrange plates May / 49

20 Theorem Inverse Problems Plates with arbitrary shapes Theorem 1 For Ω R 2 open and bounded, given x 0 R 2 \ Ω, and any 0 < T < T 0, then if u is a solution of (6), any one of the sets Γ T,G+(x 0 )(u) or ˆΓ T,G+(x 0 )(u) is enough to uniquely determine the set {f 1,..., f N } H 1 (Ω) in (6). A. Kawano (Poli-USP) Germain-Lagrange plates May / 49

21 Inverse Problems Plates with arbitrary shapes Steps in the proof Technique: Comparison with the wave equation 1. Representation of the loading 2. Representation of the solution 3. Application of the data 4. The main ingredient: Comparison with the wave equation A. Kawano (Poli-USP) Germain-Lagrange plates May / 49

22 Thanks Thank you for your attention! A. Kawano (Poli-USP) Germain-Lagrange plates May / 49

23 Thanks P is not hypo-elliptic By Hörmander s theorem, P is hypo-elliptic if and only if its symbol P satisfies There are δ, C > 0 such that P (α) (ξ) / P(ξ) C ξ α δ for ξ R m and ξ sufficiently big. The symbol is P(ξ) = (ξ1 2 + ξ2 2 )2 ξ3 2. Take α = (0, 1, 0) to see that ξ1 2 + ξ2 2 is not possible for any δ, C > 0. Back ξ 2 C ξ δ (ξ ξ 2 2 )2 ξ 2 3 A. Kawano (Poli-USP) Germain-Lagrange plates May / 49

24 Thanks Speed of signal propagation is infinite The proof is just an application of the Paley-Wiener Theorem. If the speed of propagation of signals were finite, then freezing the fundamental solution φ (1) at any fixed t > 0, the distribution x φ(t, x) would have compact support. And by the Paley-Wiener Theorem, its Fourier Transform would have controlled growth: There would exist C > 0, N 0 N and M > 0 such that φ(z, t) lim sup z + (1 + z ) N C. 0 e M Imz However, it is possible that it is false. And therefore, the speed of propagation of signals is in fact infinite. Back A. Kawano (Poli-USP) Germain-Lagrange plates May / 49

25 Unbounded plate Factorization Factor into two Schrodinger operators P 1. = ( t ı ), P 2. = ( t + ı ), 2 t = P 1 P 2. The original problem implies { P 1 w = 0, in ]0, + ) R 2, w(0) = f, (9) where w = P 2 u is supported in [0, + ). Now the set for the identification of f is Γ ]0,T [ Ω. = { w(t), ϕ : ϕ C c (Ω), t ]0, T [}, where T > 0 is arbitrarily small. Back A. Kawano (Poli-USP) Germain-Lagrange plates May / 49

26 Unbounded plate Spherical Means Result Spherical Means If φ C c (R n ), φ x denotes the function y φ(y x). Theorem 2 Suppose that Ω R n is an open connected subset, Q 1 Q 2 R and that for any t Q 2, T t D (Ω). Suppose also that there is U Ω such that T t U 0, t Q 1. Suppose that there exists a neighborhood of the origin V R n such that for all x Ω and all φ Cc (V ) such that supp(φ x ) Ω, it is true that T t, φ x = 0, t Q 1 T t, φ x+ ξr ds ξ = 0, t Q 2, (10) S 1 (0) for all R > 0 such that supp(φ x+ ξr ) Ω, ξ S 1 (0). Then T t 0, t Q 2. Back A. Kawano (Poli-USP) Germain-Lagrange plates May / 49

27 Rectangular plate Representation of the loading Representation of the loading We represent Q L 2 (R) by Q = + + m=1 n=1 Q m,n sen(mx)sen(ny). (11) Assuming that Q m,n R, m, n N, because we deal only with real valued loading. Back A. Kawano (Poli-USP) Germain-Lagrange plates May / 49

28 Rectangular plate Representation of the solution Representation of the solution We prove that the solution of the direct problem is u(t, x, y) = + + m=1 n=1 Q m,n sen(mx)sen(ny) k m,n t 0 g(t τ)sen(k m,n τ) dτ. (12) Back A. Kawano (Poli-USP) Germain-Lagrange plates May / 49

29 Application of the data Rectangular plate Application of the data After some manipulation, and invoking the data available, we have Proposition 5.1 Consider u given by (12). Let Ω R be any line segment parallel to the Ox axis that contains a point (x 0, y 0 ) R and T 0 > 0. If for any φ C c (Ω), u(t), φ = 0, t [0, T 0 ], then + + m=1 n=1 Q m,n k m,n sen(k m,n τ)sen(ny 0 ) sen(mx), φ = 0, τ [0, T 0 ]. (13) Back A. Kawano (Poli-USP) Germain-Lagrange plates May / 49

30 Rectangular plate Almost periodic distributions Almost Periodic Distributions To analyse the uniqueness conditions, we shall use some results concerning almost periodic distributions [1]. Consider a series of the form w(t) = n N a n e ı λnt, (14) where the coefficients (a n ) n N and exponents Λ = (λ n ) n N satisfy the following conditions: 1 There is a q Z + such that (n q a n ) n N l 1, that is, (a n ) n N s, the space of slowing growing sequences. 2 For Λ, we suppose that there are n 0 N, C > 0, and α > 0 such that n n 0 λ n Cn α. A. Kawano (Poli-USP) Germain-Lagrange plates May / 49

31 Rectangular plate Almost periodic distributions Definition 5.1 (Uniform discrete set) The set {d n : n N} is a uniformly discrete set if there exists δ > 0 such that i j di d j δ. (15) Definition 5.2 (upper uniform density) Given Λ = (λ n ) n N, uniformly discrete, the upper uniform density of Λ, denoted by u. u. d.(λ), is defined as u. u. d.(λ) = lim r + max x R (Λ [x, x + r]), r where (Λ [x, x + r]) denotes the cardinality of the set Λ [x, x + r]. A. Kawano (Poli-USP) Germain-Lagrange plates May / 49

32 Rectangular plate Almost periodic distributions The following result is from [1]. Theorem 3 For w of the form (14), given Λ uniformly discrete, if α > 1, then for any τ > 0, w [ τ,τ] = 0 w 0. If α = 1, Λ = O(n), then if τ > π u. u. d.(λ), then w [ τ,τ] = 0 w 0. Remember: w(t) = n N a n e ı λnt. Back A. Kawano (Poli-USP) Germain-Lagrange plates May / 49

33 Rectangular plate Application of APD to the problem Application of the uniqueness property of almost periodic distributions We order the exponents (k m,n ) non-decreasingly and call the ordered sequence Λ = ( λ ν ) ν N, we will have Λ = O(ν). Let Ω R be an arbitrary line segment parallel to the Ox axis. Defining for φ Cc (Ω), B ν (φ) = m,n N k m,n= λ ν Q m,n ( 1) ν sen(ny) sen(mx), φ (16) 2ı k m,n and Λ = (λ ν ) ν N, λ ν = ( 1) ν λ ν 2, where x is the least natural number a such that x a, equation (13) becomes B ν (φ)e ı λντ = 0, τ [0, T 0 ], φ Cc (Ω). (17) ν N A. Kawano (Poli-USP) Germain-Lagrange plates May / 49

34 Rectangular plate Application of APD to the problem Proposition 5.2 The set Λ = (λ n ) n N is uniformly discrete if and only if (L 1 /L 2 ) 2 Q. If (L 1 /L 2 ) 2 Q, then u. u. d.((λ ν ) ν N ) L 1L 2 4πh 2. Remember: L 1 and L 2 are the sides of the rectangle and h is the thickness of the plate. A. Kawano (Poli-USP) Germain-Lagrange plates May / 49

35 Rectangular plate Application of APD to the problem Theorem 4 For a given rectangular plate with sides (L 1, L 2 ) R 2 +, the set Γ a T a,ω, where Ω R is an arbitrary line segment parallel to the Ox axis that contains any point (x, y 0 ) R such that sen(ny 0 ) 0, n N, and T a > L 1L 2 4πh 2, is enough for the identification of Q L 2 (R). Proof (When L 2 1 /L2 2 Q). The proof is divided in two parts. Here we deal only with the situation when L 2 1 /L2 2 Q. From Proposition 5.2, we know that the set (λ ν ) ν N is uniformly discrete. Then by an application of Theorem 3 to equation (17), we obtain that for each φ Cc (Ω), the coefficients B ν (φ) = 0, ν N. For each ν N, we have an linear equation. Now, it suffices to use a enumerable quantity of φ s in Cc (Ω), or to take enough derivatives in the direction of the line segment Ω, to determine Q m,n, m, n N. Back A. Kawano (Poli-USP) Germain-Lagrange plates May / 49

36 Plates with arbitrary shapes Representation of the loading Engenvectors Representation of the loading Consider the eigenproblem 2 S = µ 2 S, in Ω, S(x) = 0, x Ω, u(x) = 0, x Ω, (18) for S H 1 0 (Ω) H3 (Ω). From the standard theory, we know that if S satisfies (18), then S C (Ω). The eigenvectors of (18) form an orthogonal Hilbert basis (S n ) H 1 0 (Ω). We normalize S n so that S n H 1 0 (Ω) = 1, n N. The eigenvalues are all positive and µ n = O(n). A. Kawano (Poli-USP) Germain-Lagrange plates May / 49

37 Plates with arbitrary shapes Representation of the loading Representation of the loading The loading is of the form h(t, x) = N g n (t) f n (x), j=1 f n (x) = where for each n {1,..., N}, ( bn,m + m=1 ) µ m m N b n,m S m (x), l 2. Back A. Kawano (Poli-USP) Germain-Lagrange plates May / 49

38 Plates with arbitrary shapes Representation of the solution Representation of the solution After some manipulation, the solution of the direct plate problem can be expressed as u(t, x) = t 0 [ N + g n (t τ) n=1 m=1 b n,m µ m sen(µ m τ)s m (x) ] dτ. (19) Back A. Kawano (Poli-USP) Germain-Lagrange plates May / 49

39 Plates with arbitrary shapes Application of the data Application of the data Applying data Γ T,G+(x 0 )(u) = {(t, φ, u(t, ), φ ) : t [0, T ], φ C c (VG + (x 0 ))} to the problem, we conclude that T > 0, + m=1 b n,m µ m sen(tµ m ) S m, φ = 0, t [0, T ]. (20) Taking the time derivative of the above expression, we obtain that for any n {1,..., N}, + m=1 b n,m cos(tµ m ) S m, φ = 0, t [ T ], φ C c (VG + (x 0 )). (21) Back A. Kawano (Poli-USP) Germain-Lagrange plates May / 49

40 Plates with arbitrary shapes Results for the wave equation Comparison with the wave equation For plates with arbitrary shapes, we do not have an explicit formula for the eigenvalues. To circumvent this difficulty, we look for, surprisingly, a uniqueness result for the wave equation. Consider now the wave equation problem 2 ṽ ṽ = 0, in ]0, + ) Ω, t2 ṽ = 0, at t = 0, ṽ t = Q, at t = 0, γ Ω (ṽ(t)) = 0, t 0. (22) A. Kawano (Poli-USP) Germain-Lagrange plates May / 49

41 Plates with arbitrary shapes Comparison with the wave equation For Q H 1 (Ω) expressed as Q = + m=1 a m S m, with (a m /µ m ) m N l 2, the solution of (22) is given by ṽ(t, x) = + m=1 a m µm sen( µ m t)s m (x). (23) Formally, the only difference with the one that appears in (20) the place of exponents µ m, (23) has µ m instead. is that in A. Kawano (Poli-USP) Germain-Lagrange plates May / 49

42 Plates with arbitrary shapes Comparison with the wave equation The proof of the following result for the wave equation is found in [2]. Theorem 2 For Ω R 2 convex or with boundary of class C 1,1, given x 0 R 2 \ Ω, and any T 1 > sup x Ω\G+(x 0 ) x x 0, there exists C > 0 such that T1 Q 2 H 1 (Ω) < C 0 G +(x 0 ) where ṽ and Q H 1 (Ω) are related by equation (22). ṽ(t, x) 2 dx dt, (24) Theorem 2 means that the set Γ T1,G +(x 0 )(ṽ) is enough for the unique determination of Q H 1 (Ω) in the WAVE problem (22). A. Kawano (Poli-USP) Germain-Lagrange plates May / 49

43 Plates with arbitrary shapes Comparison with the wave equation In other words, Theorem 2 states that + m=1 a m µm sen( µ m t) S m, φ = 0, t [0, T 1 ], φ C c (VG + (x 0 )) implies (a m ) m N = {0}. Equivalently, noting that the above series converges absolutely, if we take the time derivative of the above distribution, we conclude that + m=1 a m cos( µ m t) S m, φ = 0, t [0, T 1 ], φ C c (VG + (x 0 )) (a m ) m N = {0}. (25) A. Kawano (Poli-USP) Germain-Lagrange plates May / 49

44 Plates with arbitrary shapes Comparison with the wave equation Translated to the case of the Germain-Lagrange equation, our problem is to prove that (25) leads to the implication + m=1 b n,m cos(µ m t) S m, φ = 0, t [0, T t N ], φ C c (VG + (x 0 )) (b n,m ) m N = {0}, n {1,..., N}. (26) Observe that the tricky part is the absence of the square root of µ m in the expression appearing in (26). A. Kawano (Poli-USP) Germain-Lagrange plates May / 49

45 Plates with arbitrary shapes Comparison with the wave equation In what follows, we call Z τ the space of functions in S that have an extension to an entire function and whose Fourier Transforms are in C c (] τ, τ[). By the Paley Wiener Theorem, Z τ is the space of entire functions ϕ such that for all k Z +, there is a C k > 0 such that ϕ(z) C k (1 + z ) k eτ Imz, z Z. A. Kawano (Poli-USP) Germain-Lagrange plates May / 49

46 Plates with arbitrary shapes Comparison with the wave equation Now we obtain from (25) the implication: + m=1 a m ϕ( µ m t) S m, φ = 0, ϕ Z T1, φ C c (VG + (x 0 )) (a m ) m N = {0}. (27) we must prove that + m=1 b n,m ϕ(µ m t) S m, φ = 0, ϕ Z (T tn ), φ C c (VG + (x 0 )). (b n,m ) m N = {0}. (28) The implication in (27) means that the set A. = { (µ m ϕ( µ m ) S m, φ ) m N : ϕ Z T1, φ C c (VG + (x 0 )) } (29) is dense in l 2. A. Kawano (Poli-USP) Germain-Lagrange plates May / 49

47 Plates with arbitrary shapes Comparison with the wave equation Important lemma Lemma 3 Let T > 0 and (λ i ) i N C. If ϕ Z T is an entire function such that ϕ R is even, then z ϕ( z) is also an entire function and {(ϕ( λ i )) i N : ϕ Z T } { ( ϕ(λ i )) i N : ϕ Z T }, T > 0. (30) Lemma 3 and the fact that the set in (29) is dense in l 2 imply that the set { (µm ϕ(µ m ) S m, φ ) m N : ϕ Z T2, φ C c (VG + (x 0 )) } is also dense in l 2, T 2 > 0. This proves the implication in (28), (26) and Theorem 1. Back A. Kawano (Poli-USP) Germain-Lagrange plates May / 49

48 Plates with arbitrary shapes Algumas referências I Comparison with the wave equation Kawano, A. and Zine, A., Uniqueness and nonuniqueness results for a certain class of almost periodic distributions. SIAM Journal of Mathematical Analysis, n.1, V. 43, pp , Yamamoto, Masahiro and Zhang, Xu. Global Uniqueness and Stability for an Inverse Wave Source Problem for Less Regular Data. Journal of Mathematical Analysis and Applications, n.2, V.263, pp , A. Kawano (Poli-USP) Germain-Lagrange plates May / 49

49 Plates with arbitrary shapes Algumas referências II Comparison with the wave equation Stephen P. Timoshenko and S. Woinowsky-Krieger. Theory of Plates and Shells. McGraw-Hill, 2nd edition, Kawano, Alexandre. Uniqueness in the determination of vibration sources in rectangular Germain Lagrange plates using displacement measurements over line segments with arbitrary small length. Inverse Problems, n. 8, V.29, pages , Kawano, Alexandre. Uniqueness results in the identification of distributed sources over Germain-Lagrange plates by boundary measurements. Submitted, A. Kawano (Poli-USP) Germain-Lagrange plates May / 49