Pohozhaev and Morawetz Identities in Elastostatics and Elastodynamics

We construct identities of Pohozhaev type, in the context of elastostatics and elastodynamics, by using the Noetherian approach. As an application, a non-existence result for forced semi-linear isotropic and anisotropic elastic systems is established.

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Datum:	2011
Hauptverfasser:	Bozhkov, Y., Olver, P.J.
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Veröffentlicht:	Інститут математики НАН України 2011
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spelling	irk-123456789-1471762019-02-14T01:24:46Z Pohozhaev and Morawetz Identities in Elastostatics and Elastodynamics Bozhkov, Y. Olver, P.J. We construct identities of Pohozhaev type, in the context of elastostatics and elastodynamics, by using the Noetherian approach. As an application, a non-existence result for forced semi-linear isotropic and anisotropic elastic systems is established. 2011 Article Pohozhaev and Morawetz Identities in Elastostatics and Elastodynamics / Y. Bozhkov, P.J. Olver // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 31 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 35J50; 35J47; 35L51 DOI:10.3842/SIGMA.2011.055 http://dspace.nbuv.gov.ua/handle/123456789/147176 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	We construct identities of Pohozhaev type, in the context of elastostatics and elastodynamics, by using the Noetherian approach. As an application, a non-existence result for forced semi-linear isotropic and anisotropic elastic systems is established.
format	Article
author	Bozhkov, Y. Olver, P.J.
spellingShingle	Bozhkov, Y. Olver, P.J. Pohozhaev and Morawetz Identities in Elastostatics and Elastodynamics Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Bozhkov, Y. Olver, P.J.
author_sort	Bozhkov, Y.
title	Pohozhaev and Morawetz Identities in Elastostatics and Elastodynamics
title_short	Pohozhaev and Morawetz Identities in Elastostatics and Elastodynamics
title_full	Pohozhaev and Morawetz Identities in Elastostatics and Elastodynamics
title_fullStr	Pohozhaev and Morawetz Identities in Elastostatics and Elastodynamics
title_full_unstemmed	Pohozhaev and Morawetz Identities in Elastostatics and Elastodynamics
title_sort	pohozhaev and morawetz identities in elastostatics and elastodynamics
publisher	Інститут математики НАН України
publishDate	2011
url	http://dspace.nbuv.gov.ua/handle/123456789/147176
citation_txt	Pohozhaev and Morawetz Identities in Elastostatics and Elastodynamics / Y. Bozhkov, P.J. Olver // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 31 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT bozhkovy pohozhaevandmorawetzidentitiesinelastostaticsandelastodynamics AT olverpj pohozhaevandmorawetzidentitiesinelastostaticsandelastodynamics
first_indexed	2025-07-11T01:32:35Z
last_indexed	2025-07-11T01:32:35Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 7 (2011), 055, 9 pages Pohozhaev and Morawetz Identities in Elastostatics and Elastodynamics? Yuri BOZHKOV † and Peter J. OLVER ‡ † Instituto de Matemática, Estatistica e Computação Cient́ıfica - IMECC, Universidade Estadual de Campinas - UNICAMP, Rua Sérgio Buarque de Holanda, 651, 13083-859 - Campinas - SP, Brasil E-mail: bozhkov@ime.unicamp.br URL: http://www.ime.unicamp.br/~bozhkov/ ‡ School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA E-mail: olver@umn.edu URL: http://www.math.umn.edu/~olver/ Received February 01, 2011, in final form June 02, 2011; Published online June 08, 2011 doi:10.3842/SIGMA.2011.055 Abstract. We construct identities of Pohozhaev type, in the context of elastostatics and elastodynamics, by using the Noetherian approach. As an application, a non-existence result for forced semi-linear isotropic and anisotropic elastic systems is established. Key words: Pohozhaev identity; Navier’s equations; Noether’s theorem 2010 Mathematics Subject Classification: 35J50; 35J47; 35L51 1 Introduction Identities of Pohozhaev type have been widely used in the theory of partial differential equations, in particular for establishing non-existence results for large classes of forced elliptic boundary value problems and eigenvalue problems, [20, 21, 22]. The purpose of this note is to obtain and apply analogous identities in elastostatics and elastodynamics, which have not (to the authors’ knowledge) been developed to date. Our approach will be based on a fundamental identity first introduced by Noether in her seminal paper [16] that connected symmetries of variational problems to conservation laws of their Euler–Lagrange equations. As noted in [19], the identities originally due to Pohozhaev, [20, 21], owe their existence to Noether’s identity. For classical solutions of the linear equation ∆u+λu = 0 such an identity was obtained by Rellich in [24]. Further, in [25], Rellich established an integral identity for a function belonging to certain function spaces, without any reference to differential equations it may satisfy. The Rellich identity has been generalized by Mitidieri, [12, 13], for a pair of functions. General Rellich-type identities on Riemannian manifolds have been recently established in [5, 6] by use of Noether’s identity applied to conformal Killing vector fields. In [20], Pohozhaev established an integral identity for solutions of the Dirichlet problem for the semilinear Poisson equation ∆u + λf(u) = 0 in a bounded domain with homogeneous Dirichlet boundary condition. Later, for solutions of general Dirichlet problems, he obtained in [21] what is now called the Pohozhaev identity. Such identities became very popular after the paper of Pucci and Serrin, [22], where, on p. 683, the relation with the general Noetherian theory is mentioned. See also the earlier paper by Knops and Stuart, [11], and remarks in the second author’s 1986 book [19]. The Noetherian approach to Pohozhaev’s identities was further ?This paper is a contribution to the Special Issue “Symmetry, Separation, Super-integrability and Special Functions (S4)”. The full collection is available at http://www.emis.de/journals/SIGMA/S4.html mailto:bozhkov@ime.unicamp.br http://www.ime.unicamp.br/~bozhkov/ mailto:olver@umn.edu http://www.math.umn.edu/~olver/ http://dx.doi.org/10.3842/SIGMA.2011.055 http://www.emis.de/journals/SIGMA/S4.html 2 Y. Bozhkov and P.J. Olver developed and applied in [3, 4, 23, 31]. Additional applications of Rellich–Pohozhaev estimates to nonlinear elliptic theory can be found in [26, 27], while applications to nonlocal problems appear in [8]. With regard to geometric applications, [1, 7, 9, 28] develop a systematic approach to Pohozhaev-type obstructions for partial differential equations invariant under the action of a conformal group. For a relation between the Lie point symmetries of the nonlinear Poisson equation on a (pseudo-) Riemannian manifold and its isometry and conformal groups see [2]. In dynamical problems, the conformal invariance of the wave and Klein–Gordon equations was used by Morawetz, [14], to establish several very useful integral identities. These were applied by her and Strauss, [29, 30], to the study of the decay, stability, and scattering of waves in nonlinear media. In the final section, we will generalize Morawetz’ conformal identity to some dynamical systems governing waves in elastic media. Applications of our identity to decay and scattering of elastic waves will be treated elsewhere. In elastostatics, the independent variables x ∈ Rn, for n ≥ 2, represent reference body coor- dinates, while the dependent variables u = u(x) = (u1(x), . . . , un(x)) represent the deformation of the point x. The independent variable x will belong to a bounded or unbounded domain Ω ⊆ Rn that has sufficiently (piecewise) smooth boundary ∂Ω. We use ν to denote the outward unit normal on ∂Ω. For elastodynamics, we append an additional independent variable, t, repre- senting the time, and so u = u(t, x). The partial derivatives of a smooth (vector) function u(x) are denoted by subscripts: uki := ∂uk ∂xi , ukt := ∂uk ∂t , ukij := ∂2uk ∂xi∂xj , etc. The n× n spatial Jacobian matrix ∇u = (uki ) is known as the deformation gradient. We shall consistently use the Einstein summation convention over repeated indices, which always run from 1 to n. We assume that all considered functions, vector fields, tensors, functio- nals, etc. are sufficiently smooth in order that all the derivatives we write exist in the classical sense. When we say that a function is “arbitrary”, we mean that it is a sufficiently smooth function of its arguments defined on the domain Ω. Extensions of our results to more general solutions will then proceed on a case by case basis. 2 Noether’s identity A vector field v = ξi(x, u) ∂ ∂xi + φi(x, u) ∂ ∂ui on the space of independent and dependent variables induces a flow that can be interpreted as a (local) one-parameter group of transformations. The vector field is known as the infinitesimal generator of the flow, [19]. For example, the particular vector field v = axi ∂ ∂xi + b ui ∂ ∂ui , where a, b are constant, generates the group of scaling transformations (x, u) 7−→ ( λax, λbu ) . The action of the group on functions u = f(x) by transforming their graphs induces an action on their derivatives. The corresponding infinitesimal generator of the prolonged group action has the form pr(1)v = ξi(x, u) ∂ ∂xi + φi(x, u) ∂ ∂ui + φij(x, u,∇u) ∂ ∂uij , (1) Pohozhaev and Morawetz Identities in Elastostatics and Elastodynamics 3 where φij(x, u,∇u) = Djφ i − (Djξ k)uik = ∂φi ∂xj + ∂φi ∂uk ukj − ∂ξk ∂xj uik − ∂ξk ∂ul ulju i k, (2) and Dj = ∂/∂xj + ukj∂/∂u k denotes the total derivative with respect to xj . See [19] for a proof of this formula, along with its extension to higher order derivatives. For a first order Lagrangian L(x, u,∇u), Noether’s identity reads pr(1)v(L) + LDiξ i = Ei(L)(φi − uijξj) +Di [ Lξi + ∂L ∂uji (φj − ujsξs) ] , (3) where Ei is the Euler operator or variational derivative with respect to ui, [19]. Once stated, the verification of the identity is a straightforward computation. In the following sections, we will investigate how to use Noether’s identity in the framework of elasticity, and apply the corresponding integral identities to establish non-existence results. The proofs are sketched, while the full details are left to the interested reader as exercises. 3 Elastostatics We recall that the equilibrium equations for a homogeneous isotropic linearly elastic medium in the absence of body forces arise from the variational principle with Lagrangian L0(x, u,∇u) = 1 2 µ‖∇u ‖2 + 1 2 (µ+ λ)(∇ · u)2 = 1 2 µ n∑ i,j=1 ( uij )2 + 1 2 (µ+ λ) ( n∑ i=1 uii )2 , where the parameters λ and µ are the Lamé moduli. The squared norm of the deformation gradient matrix ∇u refers to the sum of the squares of its entries, while ∇ · u denotes the divergence of the deformation. The corresponding Euler–Lagrange equations are known as Navier’s equations: µ∆u+ (µ+ λ)∇(∇ · u) = 0, where the Laplacian ∆ acts component-wise on u. Henceforth, we assume that µ > 0 and µ+λ > 0, thereby ensuring strong ellipticity and positive definiteness of the underlying elasticity tensor, [10, 17]. In this paper, we shall study boundary value problems for elastic bodies that are subject to a nonlinear body-force potential F (u). Thus, we modify the preceding Lagrangian L(x, u,∇u) = 1 2 µ‖∇u ‖2 + 1 2 (µ+ λ)(∇ · u)2 − F (u), where we assume, without loss of generality, that F (0) = 0. The associated equilibrium Euler– Lagrange equations are µ∆u+ (µ+ λ)∇(∇ · u) + f(u) = 0, (4) where fi(u) = ∂F/∂ui are the components of the gradient of the body-force potential with respect to the dependent variables u. More generally, we consider Lagrangians of the form: L = 1 2 Ckl ij e i ke j l − F (u), (5) 4 Y. Bozhkov and P.J. Olver where again F (0) = 0, and e = 1 2 ( ∇u+∇uT ) , with components eik = 1 2 ( uik + uki ) , is the strain tensor. The quadratic components in the Lagrangian (5) model the stored energy of a general anisotropic linearly elastic medium, while F (u) represents a nonlinear body-force potential. The elastic moduli Ckl ij are assumed to be constant, satisfying Ckl ij = Cil kj = Ckj il = C lk ji . (6) Thus in planar elasticity there are 6 independent elastic moduli, while in three dimensions 21 independent moduli are required in general, [10]. Additional symmetry restrictions stemming from the constitutive properties of the elastic material may place additional constraints on the moduli. We may also assume Ckl ij a i ka j l ≥ 0 (7) for any matrix A = (apq). The less restrictive Legendre–Hadamard condition is that Ckl ij v ivjwkwl > 0 for any rank one matrix A = v ⊗ w. The Euler–Lagrange equations associated with (5) read Ckl ij u j kl + fi(u) = 0. (8) In general, the most basic Pohozhaev-type identity is based on the associated Noether identity for the infinitesimal generator of an adroitly chosen scaling transformation group, [19]. Theorem 1. Let Ω be a bounded domain in Rn. Then the classical solutions of (8) – that is u ∈ C2(Ω) ∩ C1(Ω̄) – subject to homogeneous Dirichlet boundary conditions on ∂Ω satisfy the following Pohozhaev-type identity:∫ Ω [ n− 2 2 ukfk(u)− nF (u) ] dx = −1 2 ∫ ∂Ω Ckl ij u i ku j l (x, ν)ds, (9) where ν is the outward unit normal to ∂Ω and (·, ·) is the Euclidean scalar product in Rn. Proof. We consider the one-parameter group of dilations (x, u) 7−→ (λx, λ(2−n)/2u) with infinitesimal generator v = xi ∂ ∂xi + 2− n 2 ui ∂ ∂ui . According to (1), (2), the first order prolongation of this vector field is pr(1)v = xi ∂ ∂xi + 2− n 2 ui ∂ ∂ui − n 2 uji ∂ ∂uji . Then one easily sees that pr(1)v(L) + LDiξ i = n− 2 2 ukfk(u)− nF (u). (10) The identity (9) now follows from the divergence theorem using (3), (5), (10), our assumption F (0) = 0, and the homogeneous Dirichlet boundary conditions, taking into account that, on ∂Ω, ujsνi = ujiνs. (11) See [22, p. 683] for more details on the last point. � Pohozhaev and Morawetz Identities in Elastostatics and Elastodynamics 5 For the sake of completeness, we specialize the general elastic Pohozhaev identity to the isotropic case of the forced Navier equations (4) in Ω:∫ Ω [ n− 2 2 ukfk(u)− nF (u) ] dx = −1 2 ∫ ∂Ω [ 1 2 µ‖∇u ‖2 + 1 2 (µ+ λ)(∇ · u)2 ] (x, ν)ds, again subject to homogeneous Dirichlet boundary conditions on ∂Ω. As a corollary, we obtain the following non-existence result. Recall that the domain Ω is star-shaped with respect to the origin if (x, ν) ≥ 0 for any x ∈ ∂Ω. Theorem 2. Suppose that Ω is a star-shaped domain. Let the function F = F (s) = F (s1, . . . , sn) ∈ C1(Rn) satisfy the conditions F (0) = 0 and n− 2 2 sk ∂F ∂sk − nF (s) ≥ 0, i = 1, . . . , n, (12) for any s ∈ Rn. We also suppose that the equality in (12) holds if and only if s = 0. Then there is no non-trivial classical solution of the potential systems (4), (8), subject to homogeneous Dirichlet boundary conditions. Proof. This theorem follows easily from the identity (9), taking into account the positivity requirement (7) and star-shapedness condition. Indeed, any classical solution of (8) subject to homogeneous Dirichlet boundary conditions on ∂Ω must satisfy the identity (9). For (7) and the star-shapedness condition (x, ν) ≥ 0 for any x ∈ ∂Ω, it follows that the right-hand side of (9) is non-positive. On the other hand, by (12) the left-hand side of (9) is positive unless u = 0 in Ω. Hence u = 0. � 4 Elastodynamics In this section, we turn our attention to hyperbolic elastodynamic systems of potential type: −uitt + Ckl ij u j kl + fi(u) = 0 (13) in R × Ω with homogeneous Dirichlet boundary conditions on R × ∂Ω. The corresponding Lagrangian is given by L = 1 2 Ckl ij e i ke j l − 1 2 uitu i t − F (u) = 1 2 Ckl ij u i ku j l − 1 2 uitu i t − F (u), (14) where the second expression follows from the requirements (6) on the elastic moduli. Theorem 3. The classical solutions of the problem (13) satisfy the following identity d dt ∫ Ω [ tE(u) + uitu i kx k + n− 1 2 uiuit ] dx = ∫ Ω [ n− 1 2 ukfk(u)− (n+ 1)F (u) ] dx + ∫ ∂Ω [ 1 2 ( Ckl ij u i ku j l + 1 2 uitu i t ) (x, ν) + t Ckl ij u i ku j l ] ds, (15) where E(u) = 1 2 ( Ckl ij e i ke j l + uitu i t ) − F (u) = 1 2 ( Ckl ij u i ku j l + uitu i t ) − F (u) is the energy density. 6 Y. Bozhkov and P.J. Olver Proof. We introduce a vector field v which is the infinitesimal generator of the dilation group (t, x, u) 7−→ ( λt, λx, λ(1−n)/2u ) . The first order prolongation of v is given by pr(1)v = t ∂ ∂t + xi ∂ ∂xi + 1− n 2 ui ∂ ∂ui − n+ 1 2 uit ∂ ∂uit − n+ 1 2 uij ∂ ∂uij . As a result, pr(1)v(L) + LDiξ i = n− 1 2 ukfk(u)− (n+ 1)F (u), (16) where the Lagrangian L is given by (14). Then, after some algebraic manipulations, the identi- ty (15) follows from the Noether identity (3) combined with (14), (11), (16), the homogeneous Dirichlet boundary conditions, and, finally, the divergence theorem. � Let Ω ⊂ Rn be a ball of radius R centered at the origin. If we assume that u(t, x) decays sufficiently rapidly as R = \|x\| → ∞, then the following conformal identity holds for the nonlinear hyperbolic system (13) in R× Rn: Corollary 1. The classical solutions of the problem (13) in R× Rn that decay rapidly at large distances satisfy the identity d dt ∫ Rn [ tE(u) + uitu i kx k + n− 1 2 uiuit ] dx = ∫ Rn [ n− 1 2 ukfk(u)− (n+ 1)F (u) ] dx. We observe that this result generalizes Morawetz’s dilational identity for nonlinear wave equations, [14, 29, 30], to elastodynamical systems. Finally, we consider a nonlinear hyperbolic system of so-called Hamiltonian type, [4], −uitt + Ckl ij u j kl +Hvi = 0, −vitt + Ckl ij v j kl +Hui = 0, (17) in R×Ω with homogeneous Dirichlet boundary conditions on R×∂Ω. (The independent variab- le xmust belong to an even dimensional space R2m.) For such systems, we obtain a generalization of Morawetz’s conformal identity [30]. Theorem 4. The classical solutions of the problem (17) satisfy the following identity d dt ∫ Ω [ tE(u, v) + ( xkujkv j t + xkvjku j t ) + n− 1 2 ( aujvjt + bvjujt )] dx = ∫ Ω [ n− 1 2 ( aukHuk + bvkHvk )] dx + ∫ ∂Ω [( Ckl ij u i kv j l + uitv i t ) (x, ν) + tCkl ij ( uitv j l νk + vjtu i kνl )] ds, (18) where the constants a and b are such that a+ b = 2 and E(u, v) = Ckl ij u i kv j l + uitv i t −H(u, v). Pohozhaev and Morawetz Identities in Elastostatics and Elastodynamics 7 Proof. In order to prove Theorem 4, we use the same scheme as in the preceding Theorem 3. Namely, we consider a vector field v which is the infinitesimal generator of the dilation group (t, x, u, v) 7−→ ( λt, λx, λa(1−n)/2u, λb(1−n)/2v ) , where the constants a and b satisfy a+ b = 2. Applying the first order prolongation pr(1)v = t ∂ ∂t + xi ∂ ∂xi + a(1− n) 2 ui ∂ ∂ui + b(1− n) 2 vi ∂ ∂vi + ( a(1− n) 2 − 1 ) uit ∂ ∂uit + ( a(1− n) 2 − 1 ) uij ∂ ∂uij + ( b(1− n) 2 − 1 ) vit ∂ ∂vit + ( b(1− n) 2 − 1 ) vij ∂ ∂vij to the Lagrangian L = 1 2 Ckl ij u i kv j l − u i tv i t −H(u, v) (19) yields pr(1)v(L) + LDiξ i = n− 1 2 ( aukHuk + bvkHvk ) , (20) when a+ b = 2. Then, after some additional work, the identity (18) follows from (19), (3), (20), (11), the homogeneous Dirichlet boundary conditions, and the divergence theorem. � Corollary 2. Let a, b and E be as in Theorem 4. Then, provided u and v decay sufficiently rapidly at large distances, d dt ∫ R2m [ tE(u, v) + ( xkujkv j t + xkvjku j t ) + n− 1 2 ( aujvjt + bvjujt )] dx = ∫ R2m [ n− 1 2 ( aukHuk + bvkHvk )] dx. Applications of these identities to the stability and scattering of waves in elastic media will be developed elsewhere. 5 Further directions We emphasize that, in order to obtain Pohozhaev and Morawetz-type identities in elastostatics and elastodynamics by the Noetherian approach developed in [3, 4], we have focussed our atten- tion on dilations, which are particular cases of conformal transformations. Further variational identities associated with other variational symmetries remain to be investigated. In particular, it would be interesting to analyze the variational identity for the semilinear Navier equations that corresponds to the first order generalized symmetry v = [ µuij + (2µ+ λ)δiju k k ] ∂ ∂uj found in [18]. In fact, the variational and (at least in three dimensions) non-variational symme- tries for isotropic linear elastostatics were completely classified in [17, 18] and the systems not only admit point symmetries, but also a number of first order generalized symmetries. In the two-dimensional case, complex variable methods, as in [15], are used to produce infinite families of symmetries and conservation laws. Also in the two-dimensional case, additional symmetries appear when 3µ+ λ = 0. In the three-dimensional case, when 7µ+ 3λ = 0, Navier’s equations admit a full conformal symmetry group, along with additional conformal-like generalized sym- metries. Although these restrictions are non-physical, they still lead to interesting divergence identities in the more general isotropic case, which can be applied to the analysis of eigenvalue problems, and also, potentially, the nonlinearly forced case. This remains to be investigated thoroughly. 8 Y. Bozhkov and P.J. Olver Acknowledgements We wish to thank the referees for their useful suggestions. Yuri Bozhkov would also like to thank FAPESP and CNPq, Brasil, for partial financial support. Peter Olver was supported in part by NSF Grant DMS 08–07317. We both would like to thank FAPESP, São Paulo, Brasil, for the grant giving Peter Olver the opportunity to visit IMECC-UNICAMP, where this work was initiated. References [1] Bourguignon J.-P., Ezin J.-P., Scalar curvature functions in a conformal class of metrics and conformal transformations, Trans. Amer. Math. Soc. 301 (1987), 723–736. [2] Bozhkov Y., Freire I.L., Special conformal groups of a Riemannian manifold and Lie point symmetries of the nonlinear Poisson equation, J. Differential Equations 249 (2010), 872–913, arXiv:0911.5292. [3] Bozhkov Y., Mitidieri E., The Noether approach to Pohozhaev’s identities, Mediterr. J. Math. 4 (2007), 383–405. [4] Bozhkov Y., Mitidieri E., Lie symmetries and criticality of semilinear differential systems, SIGMA 3 (2007), 053, 17 pages, math-ph/0703071. [5] Bozhkov Y., Mitidieri E., Conformal Killing vector fields and Rellich type identities on Riemannian mani- folds. I, in Geometric Methods in PDE’s, Lect. Notes Semin. Interdiscip. 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Pohozhaev and Morawetz Identities in Elastostatics and Elastodynamics

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