We study the asymptotic behavior of solutions with finite energy to the displacement

problem of linear elastostatics in a three-dimensional exterior Lipschitz domain

Coscia, Vincenzo

English

We study the asymptotic behavior of solutions with finite energy to the displacement problem of linear elastostatics in a three-dimensional exterior Lipschitz domain

Archivio istituzionale della ricerca - Università di Ferrara

mathematicsArticleOn the Asymptotic Behavior of D-Solutionsto the Displacement Problem of Linear Elastostaticsin Exterior DomainsVincenzo CosciaDepartment of Mathematics and Computer Science, University of Ferrara, Via Machiavelli 30, 44121 Ferrara,Italy; vincenzo.coscia@unife.itReceived: 15 November 2019; Accepted: 30 December 2019; Published: 3 January 2020Abstract: We study the asymptotic behavior of solutions with finite energy to the displacementproblem of linear elastostatics in a three-dimensional exterior Lipschitz domain.Keywords: non-homogeneous elasticity; exterior domains; existence and uniqueness theorems;asymptotic behavior; Stokes’ paradoxMSC: Primary 74B05; 35J47; 35J57; Secondary 35J201. IntroductionThe displacement problem of elastostatics in an exterior Lipschitz domain Ω of R3 consists offinding a solution to the equations [1] (Notation—Unless otherwise specified, we will use the notationof the classical monograph [1] by M.E. Gurtin. In particular, (divC[∇u])i = ∂j(Cijhk∂kuh), Lin isthe space of second–order tensors (linear maps from R3 into itself) and Sym, Skw are the spaces ofthe symmetric and skew elements of Lin respectively; if E ∈ Lin and v ∈ R3, Ev is the vector withcomponents Eijvj. BR = {x ∈ R3 : r = |x| < R}, TR = B2R \ BR, {BR = R3 \ BR and BR0 is a fixed ballcontaining ∂Ω. If f (x) and φ(r) are functions defined in a neighborhood of infinity, then f (x) = o(φ(r))means that limr→+∞( f /φ) = 0. To lighten up the notation, we do not distinguish between scalar,vector, and second–order tensor space functions; c will denote a positive constant whose numericalvalue is not essential to our purposes.)divC[∇u] = 0 in Ω,u = uˆ on ∂Ω,limR→+∞∫∂Bu(R, σ)dσ = 0,(1)where u is the (unknown) displacement field, uˆ is an (assigned) boundary displacement, B is the unitball, C ≡ [Cijhk] is the (assigned) elasticity tensor, i.e., a map from Ω× Lin→ Sym, linear on Sym andvanishing in Ω× Skw. We shall assume C to be symmetric, i.e.,E ·C[L] = L ·C[E] ∀E, L ∈ Lin, (2)and positive definite, i.e., there exists positive scalars µ0 and µe (minimum and maximum elasticmoduli [1]) such thatµ0|E|2 ≤ E ·C[E] ≤ µe|E|2, ∀ E ∈ Sym, a.e. in Ω. (3)Mathematics 2020, 8, 77; doi:10.3390/math8010077 www.mdpi.com/journal/mathematicsMathematics 2020, 8, 77 2 of 8Let D1,q(Ω), D1,q0 (Ω) (q ∈ [1,+∞)) be the completion of C∞0 (Ω) and C∞0 (Ω), respectively, withrespect to the norm ‖∇u‖Lq(Ω).We consider solutions u to equations (1) with finite Dirichlet integral (or with finite energy) thatwe call D-solutions analogous with the terminology used in viscous fluid dynamics (see [2]). Moreprecisely, we say that u ∈ D1,2(Ω) is a D-solution to equation (1)1∫Ω∇ϕ ·C[∇u] = 0, ∀ϕ ∈ D1,20 (Ω). (4)A D-solution to system (1) is a D-solution to equation (1)1, which satisfies the boundary conditionin the sense of trace in Sobolev’s spaces and tends to zero at infinity in a mean square sense [2]limR→+∞∫∂B|u(R, σ)|2dσ = 0. (5)If u is a D-solution to (1)1, then the traction field on the boundary iss(u) = C[∇u]nwhere a well defined field of W−1/2,2(∂Ω) exists and the following generalized work and energyrelation [1] holds ∫Ω∩BR∇u ·C[∇u] =∫∂Ωu · s(u),where abuse of notation∫∂Ω u · s(u) means the value of the functional s(u) ∈ W−1/2,2(∂Ω) at u ∈W1/2,2(∂Ω), and n is the unit outward (with respect to Ω) normal to ∂Ω.If uˆ ∈W1/2,2(∂Ω), denoting by u0 ∈ D1,2(Ω) an extension of uˆ in Ω vanishing outside a ball, then(1)1,2 is equivalent to finding a field u ∈ D1,20 (Ω) such that∫Ω∇ϕ ·C[∇v] = −∫Ω∇ϕ ·C[∇u0], ∀ϕ ∈ D1,20 (Ω). (6)Since the right-hand side of (6) defines a linear and continuous functional on D1,20 (Ω), and by thefirst Korn inequality (see [1] Section 13)∫Ω|∇v|2 ≤ 2µ0∫Ω∇v ·C[∇v],by the Lax–Milgram lemma, (6) has a unique solution v, and the field u = v+ u0 is a D-solution to(1)1,2. It satisfies (1)3 in the following sense (see Lemma 1)∫∂B|u(R, σ)|2dσ = o(R−1). (7)Moreover, u exhibits more regularity properties provided C, ∂Ω and uˆ are more regular. Inparticular, if C, uˆ and ∂Ω are of class C∞, then u ∈ C∞(Ω) [3].If C is constant, then existence and regularity hold under the weak assumption of strongellipticity [1], i.e.,µ0|a|2|b|2 ≤ a ·C[a⊗ b]b, ∀ a, b ∈ R3. (8)As far as we are aware, except for the property (7), little is known about the convergence at infinityof a D-solution and, in particular, whether or under what additional conditions (7) can be improved.Mathematics 2020, 8, 77 3 of 8The main purpose of this paper is just to determine reasonable conditions on C assuring that (7)can be improved.We say that C is regular at infinity if there is a constant elasticity tensor C0 such thatlim|x|→+∞C(x) = C0. (9)Let C0 and C denote the linear spaces of the D-solutions to the equationsdivC[∇h] = 0 in Ω,h = τ on ∂Ω,limR→+∞∫∂B|h(R, σ)|2dσ = 0,(10)for all τ ∈ R3 anddivC[∇h] = 0 in Ω,h = τ + Ax on ∂Ω,limR→+∞∫∂B|h(R, σ)|2dσ = 0,(11)for all τ ∈ R3, A ∈ Lin, respectively.The following theorem holds.Theorem 1. Let u be the D–solution to (1). There is q < 2 depending only on C such that∫∂B|u(R, σ)|qdσ = o(Rq−3). (12)If C is regular at infinity, then∫∂B|u(R, σ)|qdσ = o(Rq−3), ∀q ∈ (3/2,+∞), (13)and ∫∂B|u(R, σ)|qdσ = o(Rq−3), ∀q ∈ (1, 2]⇐⇒∫∂Ωuˆ · s(h) = 0, ∀h ∈ C0. (14)Moreover, if ∫∂ΩC [uˆ⊗ n] = 0,∫∂Ωuˆ · s(h) = 0, ∀h ∈ C, (15)then ∫∂B|u(R, σ)|dσ = o(R−2). (16)2. Preliminary ResultsIn this section, we collect the main tools we need to prove Theorem 1.Lemma 1. If u ∈ D1,q(Ω), for q ∈ [1, 2], then∫∂B|u(R, σ)|qdσ ≤ c(q)Rq−3∫{BR|∇u|q. (17)Mathematics 2020, 8, 77 4 of 8Moreover, if q = 2, then, for R R0,∫{BR|u|2r2≤ 4∫{BR|∇u|2. (18)Proof. Lemma 1 is well-known (see, e.g., [2,4] and [5] Chapter II). We propose a simple proof for thesake of completeness. Since D1,2(Ω) is the completion of C∞0 (Ω) with respect to the norm ‖∇u‖L2(Ω),it is sufficient to prove (17) and (18) for a regular field u vanishing outside a ball. By basic calculusand Hölder inequality,∫∂B|u(R, σ)|qdσ =∫∂B∣∣∣∣ +∞∫R∂ru(r, σ)dr∣∣∣∣qdσ = ∫∂B∣∣∣∣ +∞∫Rr2/qr−2/q∂ru(r, σ)dr∣∣∣∣qdσ≤{ ∫∂Bdσ+∞∫R|∇u(r, σ)|qr2dr}{ ∫∂B∣∣∣∣ +∞∫Rr−2q−1 dr∣∣∣∣∣q−1dσ}.Hence, (17) follows by a simple integration.From ∫{BR|u|2r2=+∞∫R∂r(r∫∂B|u(r, σ)|2dσ)− 2∫{BRur· ∂ruby Schwarz’s inequality, one gets∫{BR|u|2r2≤ 2{ ∫{BR|u|2r2∫{BR|∇u|2}1/2.Hence, (18) follows at once.Let C0 be a constant and strongly elliptic elasticity tensor. The equationdivC0[∇u] = 0 (19)admits a fundamental solution U(x − y) [6] that enjoys the same qualitative properties as thewell-known ones of homogeneous and isotropic elastostatics, defined byUij(x− y) = 18piµ(1− ν)|x− y|[(3− 4ν)δij +(xi − yi)(xj − zj)|x− y|2],where µ is the shear modulus and ν the Poisson ratio (see [1] Section 51). In particular, U(x) = O(r−1)and for f with compact support (say) the volume potentialV[ f ](x) =∫R3U(x− y) f (y)dvy (20)is a solution (in a sense depending on the regularity of f ) to the systemdivC0[∇u] + f = 0 in R3. (21)Let H1 denote the Hardy space on R3 (see [7] Chapter III). The following result is classical (see,e.g., [7]).Lemma 2. ∇2V maps boundedly Lq into itself for q ∈ (1,+∞) andH1 into itself.Mathematics 2020, 8, 77 5 of 8Lemma 3. Let u be the D-solution to (1), Then, for R R0,∫∂Ωs(u) =∫∂BRs(u), (22)and ∫∂Ωh · s(u) =∫∂Ωuˆ · s(h), ∀h ∈ C, (23)where C is the space of D-solutions to system (11).Proof. Letg(x) =0, |x| > 2R,1, |x| < R,R−1(R− |x|), R ≤ |x| ≤ 2R,(24)with R R0. Scalar multiplication of both sides of (1)1 by gh, (2) and an integration by parts yield∫∂Ωh · s(u)−∫∂Ωuˆ · s(h) = 1R∫TR(u ·C[∇h]er − h ·C[∇u]er),where er = x/r. Since R ≤ |x| ≤ 2R, by Schwarz’s inequality,∣∣∣∣∣ 1R∫TRh ·C[∇u]er∣∣∣∣∣ ≤ 2∫TRr−1|h ·C[∇u]er| ≤ c‖r−1h‖L2(TR)‖∇u‖L2(TR),∣∣∣∣∣ 1R∫TRu ·C[∇h]er∣∣∣∣∣ ≤ 2∫TRr−1|u ·C[∇h]er| ≤ c‖r−1u‖L2(TR)‖∇h‖L2(TR).Hence, letting R→ +∞ and taking into account Lemma 1, (23) follows.Lemma 4. Let u be the D-solution to (1); then, for R R0,∫∂Ω(x⊗ s(u)−C [uˆ⊗ n]) = ∫∂BR(x⊗ s(u)−C [u⊗ eR]), (25)where C is the space of D-solutions to system (11).Proof. (25) is easily obtained by integrating the identity0 = x⊗ divC[∇u] = div (x⊗C[∇u])−C[∇u]over BR and using the divergence theorem.3. Proof of Theorem 1Let ϑ(r) be a regular function, vanishing in BR and equal to 1 outside B2R, for R R0. The fieldv = ϑu is a D-solution to the equationdivC[∇v] + f = 0 in R3, (26)withf = −C[∇u]∇ϑ− divC[u⊗∇ϑ]. (27)Mathematics 2020, 8, 77 6 of 8Of course, f ∈ L2(R3) vanishes outside TR. Let C0 be a strongly elliptic elasticity tensor. Clearly,v is a D-solution to the systemdivC0[∇v] + div (C−C0)[∇v] + f = 0 in R3, (28)which coincides with u outside B2R. Since∇kV[ f ](x) = O(r−1−k), k ∈ N, ∇k = ∇ . . .∇︸ ︷︷ ︸k−times, (29)by Lemma 2, the mapw′(x) = ∇V[(C−C0)[∇w]](x) +V[ f ](x) (30)is continuous from D1,q into itself, for q ∈ (3/2,+∞). ChooseC0ijhk = µeδihδjk.Since‖∇V[(C−C0)[∇w]]‖D1,q ≤ c(q)µe − µ0µe‖w‖D1,qand [7]limq→2c(q) = 1,the map (30) is contractive in a neighborhood of 2 and its fixed point must coincide with v . Hence,there is q ∈ (1, 2) such that u ∈ D1,q(Ω) and (12) is proved.If C is regular at infinity, then by Lemma 1 and the property of ϑ,‖v′ − z′‖D1,q ≤ c(q)‖C−C0‖L∞({SR0 )‖v− z‖D1,q . (31)Since the constant c(q) is uniformly bounded in every interval [a, b] and ‖C− C0‖L∞({SR0 ) issufficiently small, u ∈ D1,q for q ∈ (3/2,+∞).Assume that ∫∂Ωuˆ · s(h) = 0, ∀h ∈ C0. (32)By Lemma 3, for R R0, ∫∂BRs(u) =∫∂BRC[∇u]eR = 0.Therefore, taking into account (27),∫R3f =∫TRf =2R∫Rϑ′(r)dr∫∂BrC[∇u]er = 0, (33)SinceV[ f ](x) =∫R3[U(x− y)−U(y)] f (y)dvy +U(x) ∫R3f ,by (33), Lagrange’s theorem and (29)∇V[ f ](x) = O(r−3),so that∇V[ f ] ∈ Lq, q ∈ (1, 2]. (34)Mathematics 2020, 8, 77 7 of 8Then, by (31), the map (30) is contractive for q in a right neighborhood of 1 so thatu ∈ D1,q(Ω), q ∈ (1, 2]. (35)Conversely, if (35) holds, then a simple computation yields∫∂Ωs(u) =∫R3C[∇u]∇g = − 1R∫TRC[∇u]er, (36)where g is the function (24). By Hölder’s inequality,1R∣∣∣∣∣∫TRC[∇u]er∣∣∣∣∣ ≤ cR{ ∫TR|∇u|3/2}2/3{ ∫TRdv}1/3≤{ ∫TR|∇u|3/2}2/3.Therefore, letting R→ +∞ in (36) yields∫∂Ωs(u) = 0and this implies (32).FromVi[ f ](x) =∫R3[Uij(x− y)−U(y)− yk∂kUij(y)]f j(y)dvy+Uij(x)∫R3f j + ∂kU(x)∫R3Uij(y) f j(y)dvyby (33), Lemma 4, Lagrange’s theorem and (29)∇V[ f ](x) = O(r−4),so that ∇V[ f ] ∈ L1. Since f ∈ L2(R3) has compact support and satisfies (33), it belongs toH1 (see [7]p. 92) and by Lemma 2 V[ f ] ∈ H1. Hence, it follows that (30) mapsH1 into itself and‖v′ − z′‖H1 ≤ ‖C−C0‖L∞({SR0 )‖v− z‖H1 .Since, by assumptions, ‖C−C0‖L∞({SR0 ) is small, (30) is a contraction and by the above argumentits (unique) fixed point must coincide with v so that ∇u ∈ L1(Ω). We aim at concluding the paper with the following remarks.(i) It is evident that the hypothesis that C is regular at infinity can be replaced by the weaker onethat |C−C0| is suitably small at a large spatial distance.(ii) The operator V maps boundedly the Hardy space Hq (q ∈ (0, 1]) into itself [7]. Hence, theargument in the proof of (16) can be used to show that ∇v ∈ Hq, q > 3/4. We can then use theSobolev–Poincaré (see [8] p. 255) to see that u ∈ Lq(Ω) for q > 1.(iii) Relation (16) is a kind of Stokes’ paradox in nonhomogeneous elastostatics: if C is regular at infinity,then the systemdivC[∇h] = 0 in Ω,h = τ on ∂Ω,∫∂Bh(R, σ)dσ= o(R−1),Mathematics 2020, 8, 77 8 of 8with τ nonzero constant vector, does not admit solutions.(iv) If C is constant and strongly elliptic, then u is analytic in Ω and at large spatial distance admitsthe representationu(x) = U(x)∫∂Ωs(u) +∇U(x)∫∂Ω(ξ ⊗ s(u)−C[uˆ⊗ n])(ξ) +ψ(x)with |x|3|ψ(x)| ≤ c. Therefore, in the homogeneous case, the conclusions of Theorem 1 holdpointwise:|x|2|u(x)| ≤ c⇐⇒∫∂Ωuˆ · s(h) = 0, ∀h ∈ C0,∫∂ΩC [uˆ⊗ n] = 0,∫∂Ωuˆ · s(h) = 0, ∀h ∈ C ⇒ |x|3|u(x)| ≤ c.Funding: This research was funded by FAR 2019 University of Ferrara.Conflicts of Interest: The author declares no conflict of interest.References1. Gurtin, M.E. The linear theory of elasticity. In Handbuch der Physik; Truesedell, C., Ed.; Springer: Berlin,Germany, 1972; Volume VIa/2.2. Galdi, G.P. An Introduction to the Mathematical Theory of the Navier—Stokes Equations. Steady—State Problems;Springer: Berlin, Germany, 2011.3. Fichera, G. Existence theorems in elasticity. In Handbuch der Physik; Truesedell, C., Ed.; Springer: Berlin,Germany, 1972; Volume VIa/2.4. Russo, R. On Stokes’ problem. In Advances in Mathematical Fluid Mechanics; Sequeira, R.R.A., Ed.; Springer:Berlin, Germany, 2010; pp. 473–512.5. Kondratev, V.A.; Oleinik, O.A. Boundary value problems for a system in elasticity theory in unboundeddomains. Korn inequalities. Uspekhi Mat. Nauk 1988, 43, 55–98. (In Russian)6. John, F. Plane Waves and Spherical Means Applied to Partial Differential Equations; Interscience:New York, NY, USA, 1955.7. Stein, E. Harmonic Analysis: Real—Variable Methods, Orthogonality, and Oscillatory Integrals; PrincetonUniversity Press: Princeton, NJ, USA, 1993.8. Coifman, R.R.; Lions, P.L.; Meyer, Y.; Semmes, S. Compensated compactness and Hardy spaces. J. Math.Pures Appl. 1993, 72, 247–286.c© 2020 by the author. Licensee MDPI, Basel, Switzerland. This article is an open accessarticle distributed under the terms and conditions of the Creative Commons Attribution(CC BY) license (http://creativecommons.org/licenses/by/4.0/).

On the Asymptotic Behavior of D-Solutions to the Displacement Problem of Linear Elastostatics in Exterior Domains

Institutional Research Information System University of Ferrara

Vincenzo Coscia

Multidisciplinary Digital Publishing Institute

https://iris.unife.it/bitstream/11392/2415387/1/Coscia%20V.%2c%20On%20the%20Asymptotic%20Behavior%20of%20D-Solutions...%2c%20Mathematics%208%281%29%20%282020%29%2c%2077.pdf

On the Asymptotic Behavior of D-Solutions to the Displacement Problem of Linear Elastostatics in Exterior Domains

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