Variational Problems for Foppl-von Karman plates
aa r X i v : . [ m a t h . O C ] F e b VARIATIONAL PROBLEMS FOR FÖPPL-VON KÁRMÁN PLATES
FRANCESCO MADDALENA, DANILO PERCIVALE, FRANCO TOMARELLI
Abstract.
Some variational problems for a Föppl-von Kármán plate subject to generalequilibrated loads are studied. The existence of global minimizers is proved under theassumption that the out-of-plane displacement fulfils homogeneous Dirichlet condition onthe whole boundary while the in-plane displacement fulfils nonhomogeneous Neumanncondition.If the Dirichlet condition is prescribed only on a subset of the boundary, then the energymay be unbounded from below over the set of admissible configurations, as shown byseveral explicit conterexamples: in these cases the analysis of critical points is addressedthrough an asymptotic development of the energy functional in a neighborhood of the flatconfiguration. By a Γ -convergence approach we show that critical points of the Föppl-von Kármán energy can be strongly approximated by uniform Palais-Smale sequences ofsuitable functionals: this property leads to identify relevant features for critical points ofapproximating functionals, e.g. buckled configurations of the plate.Eventually we perform further analysis as the plate thickness tends to , by assumingthat the plate is prestressed and the energy functional depends only on the transverse dis-placement around the given prestressed state: by this approach, first we identify suitableexponents of plate thickness for load scaling, then we show explicit asymptotic oscillatingminimizers as a mechanism to relax compressive states in an annular plate. Contents
Introduction 21. Minimization of Föppl-von Kármán functional 42. Critical points nearby a flat configuration 133. Scaling Föppl-von Kármán energy 204. Prestressed plates: oscillating versus flat equilibria. 24References 30
AMS Classification Numbers (2010):
Key Words:
Föppl-von Kármán, Calculus of Variations, Elasticity, nonlinear Neumannproblems, Monge-Ampère equation, critical points, Gamma-convergence, asymptoticanalysis, singular perturbations, mechanical instabilities.
Date : February 21, 2017. Introduction
The Föppl-von Kármán model is widely used as an effective theoretical tool in the studyof the mechanical behavior of thin elastic plates, for its ability to describe the interplaybetween membrane and bending effects (see [3]). This interplay constitutes the source ofa rich phenomenology affecting not only the macroscopic behavior but also the occurrenceof local micro-instabilities which are crucial also in the behavior of soft solids, biologicaltissues, gels ([29]). A relevant problem consists in detecting a precise geometric descriptionof such creased equilibrium configurations in dependance of the geometric and constitutiveproperties of the plate.Despite its long and controversial history, a rigorous analysis of the well posedness forvariational problems associated to the Föppl-von Kármán functional under general bound-ary conditions is still far from complete. In particular, the minimization problem undergeneral load conditions is quite subtle. The rigorous derivation of the Föppl-von Kármánplate model from three-dimensional nonlinear elasticity was proved by Friesecke, Jamesand Müller in the seminal paper [22] under the assumption of normal forces, while in [28]the authors carefully analyze the validity of such a theory under in-plane compressive forcesand study in detail the instability issue under suitable coercivity hypotheses ([28, Theorem4]).In this paper we study the existence of minimizers for the Föppl-von Kármán energy, undergeneral load conditions. In particular, we deal with Dirichlet and Neumann conditions forthe out-of-plane displacement on the whole boundary while the in-plane displacement fulfilsnonhomogeneous Neumann condition, corresponding to general assumptions on the forcesacting on the plate. The existence of minimizers is proved in several cases by exploitingthe techniques introduced in [4],[15] to circumvent the lack of coerciveness appearing inrelated nonconvex minimization problems and by taking advantage of some properties ofthe Monge-Ampère equation (see [42], [24]).We exhibit also examples where the energy of admissible configurations is not boundedfrom below, so that existence of minimizers fails and we turn our attention to the criticalpoints by performing singular perturbation analysis of the functional in a neighborhood ofa flat configuration. This analysis leads to detect critical points of the Föppl-von Kármánenergy by suitable approximations of Palais-Smale sequences associated to approximatingfunctionals. Our procedure allows to single out global buckling configurations, in caseswhen the plate has a rectangular shape. As it is well known, wrinkling type phenomenaand other micro instabilities (see [17],[20],[21],[41],[23]) manifest themselves in sheets withvery small thickness, therefore we focus our analysis on the behavior as thickness tends to and highlight the energetic competition of oscillating configurations versus flat equilibriumconfigurations.The detailed outline of the paper is as follows.In Section 1 we prove existence of minimizers for the Föppl-von Kármán energy (1.11)corresponding to a plate of prescribed thickness h > under the action of balanced loadsin three relevant cases:i) the plate is free at the boundary of a generic Lipschitz open set, while in plane uniform normal traction or mild uniform normal compression is prescribed on the whole boundary(Theorems 1.1, 1.3);ii) the plate is simply supported on the whole boundary of a strictly convex set (Theo-rem 1.6);iii) the plate is clamped on the whole boundary of a generic Lipschitz open set (Theo-rem 1.8).Moreover we focus the analysis on the cases when these conditions at the boundary areloosened, by showing explicit counterexamples where the energy is not bounded from belowand minimizers do not exist, even for balanced loads and fixed thickness h > .Section 2 is devoted to study asymptotic behavior of the energy near a flat configuration;this is achieved by scaling the out-of-plane displacements: Theorem 2.3 shows that criticalpoints of the Föppl-von Kármán energy, say weak solutions of the corresponding Euler-Lagrange equations, can be approximately reconstructed by means of uniform Palais-Smalesequences (Definition 2.2) associated to Gamma-converging simpler functionals (concern-ing Gamma-convergence and critical points we refer also to [26]). This analysis clarifiesas some relevant features of critical points, like buckled configurations related to approxi-mating energies, can be recovered by the knowledge of equilibrium configurations relatedto the flat limit problem (Examples 2.7, 2.8).In Section 3 we study the limit as h → of scaled Föppl-von Kármán energy F h when in-plane forces in (1.11) scale as f h = h α f : we show in Theorem 3.1 and Counterexample 3.3that the natural scaling of the problem (entailing convergence of energies and minimizers)occurs if α ≥ : under this restriction, if ( u h , w h ) is a minimizer of F h then the scaled pairs ( h − α u h , h − α/ w h ) provide a weakly compact sequence in H × H and the correspondingscaled energy converges to a limit energy (Theorem 3.1 and formula (3.2) therein); onthe other hand, if α ∈ [0 , then the scaled energies may be unbounded from below as h → even for free plates or simply supported or clamped ones (Counterexample 3.3 andRemark 3.4).The results obtained in Sections 1-3 lead us to examine also the case α ∈ [0 , , by studyingthe equilibrium configurations of the plate as h → through relaxation arguments appliedto an energetic functional which takes into account a prestressed state of the plate. Pre-cisely, in Section 4: we perform the analysis of corresponding asymptotic minimizers, showa competition between oscillating and flat equilibria and highlight how this competition isruled by the mechanical and geometrical parameters: oscillating equilibria act as a mech-anism to release compression states in the limit.Eventually we exhibit a list of creased and non creased equilibrium configurations of anannular plate (Examples 4.5 -4.8), together with a general strategy (Remark 4.9) to buildthese examples: if both eigenvalues in the stress tensor of the prestressed state are strictlypositive almost everywhere, then we can expect only the flat minimizer; whereas possibleoccurrence of oscillating configurations requires the presence of a compressive state on aregion of positive measure (Proposition 4.3, Remark 4.4).The issues involved in the present article are closely related with a large class of instabili-ties, according to recent studies ([7], [8], [9], [11], [12],[10], [17], [30], [31], [32], [41]). Notation . Sym , ( R ) denotes × real symmetric matrices; a ⊗ b denotes the matrix withentries a i b j , a ⊙ b = ( a ⊗ b + a ⊗ b ) and | a | = P i a i for every a , b ∈ R n ; moreover | A | = P i,j A ij and A : B = P i,j A ij B ij , for every A , B ∈ Sym , ( R ) with entries respectively A ij , B ij . H k (Ω) denotes the Sobolev space of functions in the open set Ω ⊂ R whose distributionalderivatives up to the order k belong to L (Ω) ; H k (Ω) denotes the completion of compactlysupported functions in the Sobolev H k norm; H (Ω , R ) denotes the vector fields withcomponents in H (Ω) . − R A v d x = | A | − R A v d x ∀ measurable set A and every integrable function v defined on A . A ( x ) = 1 if x ∈ A , A ( x ) = 0 if x A . χ U ( v ) = 0 if v ∈ U , χ U ( v ) = + ∞ if v U .1. Minimization of Föppl-von Kármán functional
Let Ω ⊂ R be a bounded open connected set with Lipschitz boundary ∂ Ω , x = ( x , x ) denotes the coordinates of points in Ω referring to the canonical reference frame in R and s > is the thickness of a thin plate-like region whose reference configuration is Ω × ( − s , s ) ; moreover set s := hs where h is an non-dimensional scale factor which re-mains fixed throughout this Section.Let u : Ω → R and w : Ω → R be respectively the in-plane and out-of-plane displacements.In the geometrical linear setting the stretching tensor D is given by(1.1) D ( u , w ) = E ( u ) + 12 Dw ⊗ Dw, where(1.2) E ( u ) = 12 ( D u + D u T ) denotes the linearized strain tensor.The kernel of E , that is the set of infinitesimal rigid displacements in Ω , is denoted by(1.3) R := { u : E ( u ) = } and R ( u ) denotes the projection of u ∈ H (Ω , R ) on R .The elastic energy of a plate of thickness hs > is the sum of a membrane energy(1.4) F mh ( u , w ) = hs Z Ω J ( D ( u , w )) d x and a bending energy(1.5) F bh ( w ) = h s Z Ω J ( D w ) d x . We assume that for every A ∈ Sym , ( R ) the energy density J is given by(1.6) J ( A ) = E − ν ) (cid:0) | Tr( A ) | − − ν )det A (cid:1) = E ν ) | A | + E ν − ν ) | Tr A | where E > is the Young modulus and ν is the Poisson ratio, − < ν < / .A straightforward consequence of (1.6) which will be exploited in subsequent computationsis(1.7) c ν E | A | ≤ J ( A ) ≤ C ν E | A | where < c ν := min { (1 − ν ) − , (1+ ν ) − } ≤ C ν := max { (1 − ν ) − , (1+ ν ) − } < + ∞ .By denoting the unit outer normal to ∂ Ω by n , we define(1.8) A := { w ∈ H (Ω) | w = ∂w∂ n = 0 on Γ }A := { w ∈ H (Ω) | w = 0 on Γ }A := H (Ω) where the spaces A = A (Γ) , A = A (Γ) actually depend on Γ . We assume in generalthat(1.9) Γ ⊂ ∂ Ω is a Borel set s.t. H (Γ) > . Let(1.10) f h ∈ L ( ∂ Ω , R ) , g h ∈ L (Ω) respectively be the densities of a given in-plane load distribution and of a given out-ofplane load distribution.By taking into account the work of external loads and different types of boundary condi-tions, we define the Föppl-von Kármán functional , shortly denoted by
FvK in the sequel,(1.11) F h ( u , w ) == hs Z Ω J ( D ( u , w )) d x + h s Z Ω J ( D w ) d x − hs Z Ω g h w d x − hs Z ∂ Ω f h · u d H . Throughout the paper we choose units of measurement such that s = 1 .Equilibrium configurations of the plate under prescribed loads f h and g h are obtained byminimizing the functional (1.11) over H (Ω , R ) ×A i , i = 0 , , , corresponding respectivelyto clamped, simply supported and free plate. The present Section focuses on issues relatedto existence and non existence of these minimizers: we study in detail existence of suchminimizers according to the various choices i = 0 , , of boundary conditions and loadsand we exhibit some counterexamples in which the functional is unbounded from below,hence global minimizers do not exist.The main obstruction in applying the direct methods of the calculus of variations to thisproblem relies in the possible lack of coerciveness of the functional (1.11): indeed the kernelof the membrane energy density, which in general is a subset of the set of solutions of the Monge-Ampère equation in Ω (see Lemma 1.5 below), may be too large to allow balancingof the internal membrane energy versus the effect of external forces, in order to achievean equilibrium configuration. Notwithstanding this difficulty, an existence theorem can beproved either assuming a sign condition on boundary forces, or an homogeneous Dirichletcondition on the transverse displacement. In the first case the work of the external forces is bounded away from zero on the kernel of the membrane energy density, thus allowingthe global energy to be bounded from below; in the second one a uniqueness result in thetheory of Monge-Ampère equation implies that the kernel of bending energy reduces tothe null transverse displacement (see also [30], [31], [32]). These settings together witha tuning of some techniques introduced in [4] and [15] yield compactness of minimizingsequences, hence existence of minimizers via the direct method.Assuming f h = f h n , we prove existence of minimizers for F h in H (Ω , R ) × H (Ω) , firstunder the assumption that f h is a nonnegative constant (Theorem 1.1), second under theassumption that f h is a small negative constant (Theorem 1.3). Theorem 1.1. ( uniform boundary traction of a free plate )Assume that Ω ⊂ R is a bounded connected Lipschitz open set and (1.12) Z Ω g h d x = Z Ω x g h d x = Z Ω x g h d x = 0 , (1.13) f h = f h n on ∂ Ω , f h ≥ is a constant.Then, for every fixed h > , F h achieves a minimum over H (Ω , R ) × H (Ω) .Proof. In order to achieve the proof it will be enough to show a minimizing sequence equi-bounded in H (Ω , R ) × H (Ω) , since F h is sequentially l.s.c. with respect the weak con-vergence in such space. Due to inf H × H F h ≤ F h ( , ≤ , if F h ( u n , w n ) → inf H × H F h we may suppose F h ( u n , w n ) ≤ so, by Divergence Theorem, (1.13) and (1.7) we also get(1.14) c ν h E Z Ω | D w n | + c ν h E Z Ω | D ( u n , w n ) | ≤ hf h Z Ω div u n + h Z Ω g h w n + 1 . Set λ n := k E ( u n ) k L and suppose by contradiction that sup λ n = + ∞ , hence (up to subse-quences without relabeling) λ n → + ∞ . Let ζ n := λ − / n w n , v n := λ − n u n and x Ω is thecenter of mass of Ω . Possibly different constants denoted by C actually depend only on Ω . Then by substituting in (1.14) and dividing times λ n , we get via (1.12) and Poincarèinequality(1.15) c ν h E Z Ω | D ζ n | + λ n c ν h E Z Ω | D ( v n , ζ n ) | ≤≤ hf h Z Ω div v n + λ − / n h Z Ω g h ζ n + λ − n == hf h Z Ω div v n + λ − / n h Z Ω g h (cid:16) ζ n − − Z Ω ζ n − ( x − x Ω ) − Z Ω Dζ n (cid:17) + λ − n ≤≤ hf h Z Ω div v n + λ − / n h k g h k L + λ − / n C Z Ω | D ζ n | + λ − n . The above inequality together with k E ( v n ) k L = 1 entail(1.16) c ν h E Z Ω | D ζ n | + λ n c ν h E Z Ω | D ( v n , ζ n ) | ≤ C for large n . Exploiting k E ( v n ) k L = 1 , once more, we get Dζ n are then equibounded in H (Ω , R ) , and, up to subsequences, ζ n − − R Ω ζ n → ζ weakly in H (Ω) , Dζ n → Dζ in L (Ω , R ) due to Rellich Theorem and v n → v weakly in H (Ω , R ) .By taking into account (1.12) we get(1.17) hf h Z Ω div v n + λ − / n h Z Ω g h ζ n = hf h Z Ω div v n + λ − / n h Z Ω g h (cid:0) ζ n − − Z Ω ζ n (cid:1) → hf h Z Ω div v . By sequential lower semicontinuity together with (1.17), (1.15) we get(1.18) c ν h E Z Ω | D ζ | ≤ lim inf c ν h E Z Ω | D ζ n | ≤≤ lim inf (cid:26) hf h Z Ω div v n + λ − / n h Z Ω g h (cid:0) ζ n − − Z Ω ζ n (cid:1) + λ − n (cid:27) = hf h Z Ω div v . Moreover, by taking into account that λ n → + ∞ ,(1.19) λ n c ν h E Z Ω | D ( v n , ζ n ) | ≤ hf h Z Ω div v n + λ − n + λ − / n h Z Ω g h (cid:0) ζ n − − Z Ω ζ n (cid:1) ≤ C and by Dζ n → Dζ in L (Ω , R ) , we have also(1.20) c ν h E Z Ω | D ( v , ζ ) | ≤ lim inf c ν h E Z Ω | D ( v n , ζ n ) | ≤ C lim inf λ − n = 0 . Hence D ( v n , ζ n ) → D ( v , ζ ) = 0 , E ( v n ) → E ( v ) both in L (Ω , Sym , ( R )) and div v = −| Dζ | .Therefore by (1.18)(1.21) c ν h E Z Ω | D ζ | + 12 hf h Z Ω | Dζ | ≤ and by taking into account that − R Ω ζ = 0 we get ζ = 0 and E ( v ) = 0 , a contradiction since k E ( v n ) k L = 1 and E ( v n ) → E ( v ) in L (Ω , Sym , ( R )) . So λ n ≤ C for some C > and u n − R ( u n ) are equibounded in H (Ω , R ) by Korn inequality, while equiboundedness of w n − − R Ω w n in H (Ω) follows from (1.14). Existence of minimizers is then straightforwardvia direct method. (cid:3) If f < then the analogous of Theorem 1.1 for in-plane compression along the wholeboundary cannot be true, as shown by the next particularly telling Counterexample 1.2.Anyway we can deal also with load corresponding to small negative f , as shown by Theorem1.3 below. Counterexample 1.2. (uniform boundary compression) . Assume(1.22)
Ω = ( − , × ( − , , Γ = {− } × [ − , , g h ≡ (1.23) f h = f h n on ∂ Ω , where f h is a given constant s.t. f h < − C ν E h . Then inf F h = −∞ over both H (Ω , R ) × A and H (Ω , R ) × A .Indeed, let u = − (2 + x ) e , ϕ = (2 + x ) , and u n := n u , ϕ n := √ nϕ ; then E ( u n ) = − Dϕ n ⊗ Dϕ n and by (1.7) F h ( u n , ϕ n ) ≤ h C ν nE Z Ω | D ϕ | d x − nh f h Z ∂ Ω n · u d H == h C ν nE Z Ω | D ϕ | d x − nh f h Z Ω div u d x == h C ν nE Z Ω | D ϕ | d x + nhf h Z Ω | Dϕ | d x = nh C ν (cid:0) h E + 64 f h C ν − (cid:1) → −∞ . Referring to the bounded connected Lipschitz open set Ω ⊂ R , denote by K (Ω) the bestconstant such that(1.24) Z Ω (cid:12)(cid:12)(cid:12)(cid:12) v − − Z Ω v (cid:12)(cid:12)(cid:12)(cid:12) d x ≤ K (Ω) Z Ω | D v | d x ∀ v ∈ H (Ω , R ) . Theorem 1.3. (mild uniform boundary compression of a simply supported plate) .Assume that Ω ⊂ R is a bounded connected Lipschitz open set and (1.25) f h = f h n on ∂ Ω where f h is a given constant such that (1.26) f h > − h c ν E K (Ω) . Then, for every fixed h > , F h achieves a minimum over H (Ω , R ) × H (Ω) ∩ H (Ω) .Proof. Here, by setting
Γ = ∂ Ω , we have A = H (Ω) ∩ H (Ω) . Let F h ( u n , w n ) → inf H ×A F h and assuming by contradiction that k E ( u n ) k → + ∞ . By arguing as in theproof of Theorem 1.1 we can build a sequence ( v n , ζ n ) → ( v , ζ ) weakly in H (Ω , R ) × H (Ω) , k E ( v n ) k = 1 , D ( v n , ζ n ) → D ( v , ζ ) = O , E ( v n ) → E ( v ) both in L (Ω , Sym , ( R )) , div v = −| Dζ | and(1.27) c ν h E Z Ω | D ζ | + 12 hf h Z Ω | Dζ | ≤ we emphasize that ζ n = 0 at ∂ Ω entails − R Ω Dζ n = 0 , therefore | R Ω g h ζ n | ≤ C k g h k L k D ζ n k L for a suitable constant C = C (Ω) ; hence (1.27) can be achieved even without assuming (1.12).Therefore by taking into account that R Ω Dζ = 0 (due to ζ ∈ H ), Poincarè inequality(1.24) and assumption (1.26) altogether entail(1.28) c ν h E K (Ω) Z Ω | Dζ | + 12 hf h Z Ω | Dζ | ≤ c ν h E Z Ω | D ζ | + 12 hf h Z Ω | Dζ | ≤ , So Dζ = 0 and, by D ( v , ζ ) = O , E ( v ) = O , that is a contradiction since k E ( v n ) k L = 1 and E ( v n ) → E ( v ) in L (Ω , Sym , ( R )) . The claim follows by repeating last part of Theorem1.1 proof: here transverse load balancing (1.12) is not needed, due to boundary condition A . (cid:3) Remark 1.4.
By inspection of the proof of Theorem 1.3 we deduce also existence theoremsfor a plate clamped on a possibly proper subset Γ of the boundary. Precisely, assuming Ω bounded, connected, Lipschitz, (1.9), (1.25) with f h > − ( h c ν E ) / (12 e K (Ω , Γ)) , where e K (Ω , Γ) is the best constant s.t. R Ω (cid:12)(cid:12) v | dx ≤ K (Ω , Γ)) (cid:8)R Ω | D v | dx + R Γ (cid:12)(cid:12) v | d H (cid:9) , then F h achieves a minimum over H (Ω , R ) × A (Γ) .Similar claims in H (Ω , R ) ×A (Γ) (for plates supported on Γ ) fail, even by adding assump-tion R Ω x g h d x = R Ω x g h d x = 0 . Indeed, if Ω = (0 , , Γ = { }× [0 , , g h ≡ , f h = − λ h n ,then inf F h = −∞ , as shown by u = − (1 / x + m ) e , w m = (cid:0) ( x + m ) − m (cid:1) / , m ∈ N . Concerning existence of minimizers for F h in H (Ω , R ) × A i for i = 0 , , when Γ = ∂ Ω ,that is for clamped and simply supported plates respectively at the whole boundary, inpresence of boundary forces which fulfils neither condition (1.13) nor conditions (1.25)-(1.26) we need to state first the following Lemma (see also [22, Proposition 9]) whichclarifies the link between ker D and the solutions of the Monge-Ampère equation in Ω . Lemma 1.5.
Let Ω ⊂ R be an open set and assume that u ∈ H (Ω , R ) , ϕ ∈ H (Ω) satisfy E ( u ) + Dϕ ⊗ Dϕ = 0 in Ω .Then det D ϕ ≡ in Ω , where det D ϕ is the pointwise hessian of ϕ .Proof. Since E ( u ) satisfies the compatibility equation E , + E , = 2 E , in the sense of D ′ (Ω) , we get Z Ω ψ , ( E , − E , ) + ψ , ( E , − E , ) d x = 0 , ∀ ψ ∈ C ∞ (Ω) . Therefore since Dϕ ⊗ Dϕ = − E ( u ) we get E , = − ϕ , ϕ , E , = − ϕ , ϕ , − ϕ , ϕ , E , = − ϕ , ϕ , E , = − ϕ , ϕ , − ϕ , ϕ , . Summarizing Z Ω ψ , ( ϕ , ϕ , − ϕ , ϕ , ) + ψ , ( ϕ , ϕ , − ϕ , ϕ , ) d x = 0 , ∀ ψ ∈ C ∞ (Ω) . that is Det D ϕ = 0 where Det D ϕ is the distributional hessian of ϕ .Since ϕ ∈ H (Ω) we have det D ϕ = Det D ϕ = 0 in Ω . (cid:3) We are now in a position to state and prove an existence theorem for simply supportedplates, whose proof relies on a result by Rauch & Taylor (see [42, Theorem 5.1]) about theDirichlet problem for the
Monge-Ampère equation (see also [24]).
Theorem 1.6. ( simply supported plate )If Ω ⊂ R is bounded strictly convex and f h is an equilibrated in-plane load distribution,say (1.29) Z ∂ Ω f h · z d H = 0 ∀ z ∈ R . Then, for every fixed h > , the FvK functional F h in (1.11) achieves a minimum over H (Ω , R ) × H (Ω) ∩ H (Ω) .Proof. Here Γ ≡ ∂ Ω so, referring to (1.8), we look for minimizers of F h over H (Ω , R ) ×A = H (Ω , R ) × H (Ω) ∩ H (Ω) . The proof will be achieved by showing the existence of aminimizing sequence equibounded in H (Ω , R ) × H (Ω) , since F h is sequentially l.s.c.with respect the weak convergence in this space. Due to inf H ×A F h ≤ F h ( , ≤ ,hence if F h ( u n , w n ) → inf H ×A F h we may suppose F h ( u n , w n ) ≤ . So by taking intoaccount (1.29) and (1.7) we get via Korn and Poincarè inequality(1.30) c ν h E Z Ω | D w n | + c ν h E Z Ω | D ( u n , w n ) | ≤ h Z Ω f h · u n + h Z Ω g h w n + 1 == h Z Ω f h · (cid:0) u n −R ( u n ) (cid:1) + h Z Ω g h w n + 1 ≤ k E ( u n ) k L k f h k L + h k g h k L k Dw n k L +1 . Set λ n := k E ( u n ) k L , assume by contradiction λ n → + ∞ and set v n := λ − n u n ζ n := λ − / n w n . By substituting in (1.30) and dividing times λ n , via Poincarè inequality in H ∩ H , we get(1.31) c ν h E Z Ω | D ζ n | + λ n c ν h E Z Ω | D ( v n , ζ n ) | ≤≤ k f h k L + λ − / n h k g h k L k Dζ n k L + λ − n ≤≤ C + λ − / n h Z Ω | Dζ n | ≤ C + λ − / n h Z Ω | D ζ n | thus obtaining as in the proof of Theorem 1.1(1.32) c ν h E Z Ω | D ζ n | + λ n c ν h E Z Ω | D ( v n , ζ n ) | ≤ C ′ for a suitable C ′ > . Since k E ( v n ) k L = 1 , Dζ n are then equibounded in H (Ω , R ) so,up to subsequences, ζ n → ζ weakly in H (Ω) , Dζ n → Dζ strongly in L (Ω , R ) , v n → v weakly in H (Ω , R ) and D ( v n , ζ n ) → strongly in L (Ω) . Hence(1.33) E ( v n ) + Dζ n ⊗ Dζ n → E ( v ) + Dζ ⊗ Dζ = O strongly in L (Ω , Sym , ( R )) and E ( v n ) → E ( v ) strongly in L (Ω , Sym , ( R )) . Then by Lemma 1.5 we have det D ζ = 0 and by taking into account that Ω is strictly convex and ζ = 0 on the whole ∂ Ω byuniqueness Theorem 5.1 in [42] we get ζ ≡ in Ω . This implies E ( v ) = − Dζ ⊗ Dζ = O ,which is a contradiction since k E ( v n ) k L = 1 . Hence λ n ≤ C for suitable C > , so u n − R ( u n ) are equibounded in H (Ω , R ) and equiboundedness of w n in H (Ω) followsfrom (1.32). Existence of minimizers is obtained via direct method. (cid:3) Existence of minimizers may fail when Γ ∂ Ω even if the in-plane load f h is equilibrated,as shown by the next Counterexample. Counterexample 1.7. ( buckling under in-plane shear ) Fix γ > , ε > , h < γ/ (6 EC ν ) and Ω ε = { ( x , x ) : | x | < ε (1 − x ) , | x | < ε (4 − x ) } , (1.34) Γ ε = ∂ Ω ε ∩ { ( x , x ) : | x − x | ≥ } , (1.35) f h := γ τ ( Σ , ± − Σ , ± ) , where τ denotes the counterclockwise oriented unit vector tangent to ∂ Ω ε = Σ , ± ε ∪ Σ , ± ε and Σ , ± ε = { ( x , x ) : | x | ≤ , x = ± (1 + ε (4 − x )) } Σ , ± ε = { ( x , x ) : | x | ≤ , x = ± (2 + ε (1 − x )) } . We claim that there exists e ε such that inf F h = −∞ over H (Ω e ε , R ) × A under theassumptions listed above, notwithstanding the strict convexity of Ω e ε and the fact thatcondition (1.29) holds true.Indeed, let ψ ∈ C , ( R ) be an even function, with spt ψ ⊂ [ − , , ψ ′ = − in [1 / , / and | ψ ′′ | ≤ in R . We set ϕ ( x , x ) = ψ ( x − x ) and define w n := √ nϕ and u n := n u ,where u ( x , x ) = − u ( x , x ) = 12 Z x − x − | ψ ′ ( τ ) | dτ . By setting Ω := ( − , × ( − , ⊂ Ω ε , there is C > such that for every < ε ≤ (cid:12)(cid:12)(cid:12)(cid:12) Z ∂ Ω ε f h · u d H − Z ∂ Ω f h · u d H (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ε , hence by (1.35) and there exists e ε ∈ (0 , such that(1.36) Z ∂ Ω e ε f h · u d H ≥ Z ∂ Ω f h · u d H − γ γ Z Ω E ( u ) d x − γ . So u , ( x , x ) = − | ψ ′ ( x − x ) | = − ϕ , , u , ( x , x ) = − | ψ ′ ( x − x ) | = − ϕ , ,u , + u , (cid:20) | ψ ′ ( x − x ) | + 12 | ψ ′ ( x − x ) | (cid:21) = 12 | ψ ′ ( x − x ) | = − ϕ , ϕ , that is E ( u n ) = − Dw n ⊗ Dw n and moreover, by (1.7), (1.36) and ϕ, = − ϕ, we deduce(1.37) F h ( u n , w n ) ≤ C ν h nE Z Ω | D ϕ | dx + C ν h nE Z Ω e ε \ Ω | D ϕ | d x ++ hnγ Z Ω ϕ , ϕ , d x + h γ ≤≤ C ν h nE (cid:16)(cid:12)(cid:12) { ( x , x ) ∈ Ω : 4 | x − x | ≤ or ≤ | x − x | ≤ } (cid:12)(cid:12) + | Ω e ε \ Ω | (cid:17) + − hnγ |{ ( x , x ) ∈ Ω : 1 ≤ | x − x | ≤ }| + hn γ ≤≤ C ν Eh n − hn γ → −∞ as n → + ∞ whenever EC ν h < γ thus proving the claim. (cid:3) Clearly Theorems 1.1, 1.3, 1.6 hold for the clamped plate too: minimization in H (Ω , R ) ×A . Even better, in the case of clamped plate we can drop both convexity assumption on Ω and equilibrated out-of-plane load (1.12) as it is shown by the next result. Theorem 1.8. ( clamped plate )If Ω is a bounded connected Lipschitz open set and (1.29) holds, then for every fixed h > the functional F h in (1.11) achieves its minimum over H (Ω , R ) × H (Ω) .Proof. Again we need only to exhibit an equibounded minimizing sequence. Indeed, asin the proof of Theorem 1.6 if F h ( u n , w n ) → inf H × H F h we may suppose F h ( u n , w n ) ≤ . Then, since Γ = ∂ Ω entails H (Ω) = A ⊂ A , by setting λ n := k E ( u n ) k L , v n := λ − n u n , ζ n := λ − / n w n and assuming λ n → + ∞ , arguing as in the previous proofs weachieve the estimates (1.30), (1.31), (1.32). Then the sequence Dζ n is equibounded in H (Ω , R ) so, up to subsequences, ζ n → ζ weakly in H (Ω) , Dζ n → Dζ in L (Ω , R ) , v n → v weakly in H (Ω , R ) and D ( v n , ζ n ) → O in L (Ω , Sym , ( R )) . Hence(1.38) E ( v n ) + Dζ n ⊗ Dζ n → E ( v ) + Dζ ⊗ Dζ = O strongly in L (Ω , Sym , ( R )) , E ( v n ) → E ( v ) strongly in L (Ω , Sym , ( R )) and by Lemma 1.5 we have det D ζ = 0 in thewhole Ω . Since ζ ≡ ∂ζ∂ n ≡ on ∂ Ω , there exists a disk e Ω (bounded and strictly convex!) suchthat Ω ⊂ e Ω and the trivial extension e ζ of ζ in e Ω belongs to H ( e Ω) . Therefore det D e ζ = 0 on e Ω and still by Theorem 5.1 in [42] we get e ζ ≡ in e Ω hence ζ ≡ in Ω . Then by (1.38) E ( v ) = O , a contradiction since k E ( v n ) k L = 1 . (cid:3) Critical points nearby a flat configuration
When existence of global minimizers fails because the energy is unbounded from below, it isnatural to investigate the structure of local minimizers or, more in general of critical points.Since the nonlinearity in the
FvK functional relies in the interaction between membraneand bending contributions, we will focus in this section on the asymptotic analysis ofcritical points in the neighborhood of a flat configuration, i.e. we will study the behaviorfor small out-of-plane displacements. Throughout this section we assume that h > is fixedand(2.1) g h ≡ that is, we restrict our analysis to the case of in-plane load acting on a plate of prescribedthickness. Assume f h ∈ L ( ∂ Ω , R ) and (1.29) holds true. For every ( u , w ) ∈ H (Ω , R ) × H (Ω) , referring to (1.1) - (1.11), we enclose boundary conditions in the functional, bysetting(2.2) F ih ( u , w ) = F h ( u , w ) if u ∈ H (Ω , R ) , w ∈ A i , + ∞ otherwise ,(2.3) F ih,ε ( u , w ) = F ih ( u , εw ) , ∀ ε > . By noticing that F h, := F ih, actually is independent of i , we also set(2.4) E ih,ε ( u , w ) = ε − (cid:18) F ih,ε ( u , w ) − min H (Ω , R ) F h, (cid:19) , (2.5) E ih ( u , w ) = F bh ( w ) + h Z Ω J ′ ( E ( u )) : Dw ⊗ Dw d x if ( u , w ) ∈ { argmin F h, } × A i + ∞ else in H (Ω , R ) × H (Ω) where(2.6) J ′ ( A ) = E ν A + Eν − ν (Tr A ) I denotes the derivative of J .Functionals E ih,ε and F ih,ε are linked via the following result Proposition 2.1. E ih = Γ lim ε → + E ih,ε .Precisely, the following relations hold true:i) for every ( u ε , w ε ) ⇀ ( u , w ) in w − H × H we have (2.7) lim inf ε → E ih,ε ( u ε , w ε ) ≥ E ih ( u , w ); ii) for every ( u , w ) ∈ H × H there exists ( e u ε , e w ε ) ⇀ ( u , w ) in w − H × H such that (2.8) lim ε → E ih,ε ( e u ε , e w ε ) = E ih ( u , w ) . Proof.
Let ( u ε , w ε ) ⇀ ( u , w ) in w − H × H : by convexity we have(2.9) F ih,ε ( u ε , w ε ) ≥ ε F bh ( w ) + h R Ω J ( E ( u ε )) dx ++ hε R Ω J ′ ( E ( u ε )) : Dw ε ⊗ Dw ε dx − h R ∂ Ω f h · u ε d H ≥≥ ε F bh ( w ) + hε R Ω J ′ ( E ( u ε )) : Dw ε ⊗ Dw ε dx + min F h, and by taking into account that Dw ε ⊗ Dw ε → Dw ⊗ Dw strongly in L (Ω , Sym , ( R )) and J ′ ( E ( u ε )) ⇀ J ′ ( E ( u )) weakly in L (Ω , Sym , ( R )) , we get lim inf ε → E ih,ε ( u ε , w ε ) ≥ E ih ( u , w ) and i) is proven. The proof of ii) is achieved by taking ( e u ε , e w ε ) ≡ ( u , w ) . (cid:3) We recall that if I : X → R is any C functional defined on a Banach space X then x ∈ X is a critical point for I if I ′ ( x ) = 0 where I ′ : X → X ∗ denotes the Gateaux differential of I .Due to formula (2.10) below, F ih,ε is a C functional in the Hilbert space H (Ω , R ) × A i :precisely, for every ( u , w ) ∈ H (Ω , R ) × A i the Gateaux differential of F ih,ε at ( u , w ) isgiven by ( F ih,ε ) ′ ( u , w )[( z , ω )] = (cid:16) τ ( u , w )[ z ] , τ ( u , w )[ ω ] (cid:17) , ∀ z ∈ H (Ω , R ) , ∀ ω ∈ A i , where(2.10) τ ( u , w )[ z ] := h Z Ω J ′ (cid:18) E ( u ) + ε Dw ⊗ Dw (cid:19) : E ( z ) − h Z ∂ Ω f h · z ,τ ( u , w )[ ω ] := ε h Z Ω J ′ ( D w ) : D ω + ε h Z Ω J ′ (cid:18) E ( u ) + ε Dw ⊗ Dw (cid:19) : Dw ⊙ Dω . (cid:0) τ ( u , w )[ z ] , τ ( u , w )[ ω ] (cid:1) is replaced by the shorter notation (cid:0) τ [ z ] , τ [ ω ] (cid:1) , whenever thedependance on fixed choice for ( u , w ) is understood. Actually (2.10) provides the explicitinformation that ( F ih,ε ) ′ ( u , w ) depends continuously on ( u , w ) .Hence the Föppl-von Karman plate equations in weak form together with boundary con-ditions can be written as follows:(2.11) u , w ∈ H (Ω , R ) × A i ,τ ( u , w )[ z ] = 0 ∀ z ∈ H (Ω , R ) ,τ ( u , w )[ ω ] = 0 ∀ ω ∈ A i . Clearly ( E ih,ε ) ′ ( u , w ) = ε − ( F ih,ε ) ′ ( u , w ) hence F ih,ε and E ih,ε have the same critical points.Moreover if u ∗ ∈ argmin F h, then τ ( u ∗ , ≡ and ( u ∗ , is a critical point for F ih,ε .The next definition tunes the standard notion of Palais-Smale sequence to the presentcontext.
Definition 2.2.
Let I ε : X → R be a sequence of C functionals and X be a Banach space X . A sequence { x ε } ⊂ X is a uniform Palais-Smale sequence if there exists C > such that I ε ( x ε ) ≤ C and k I ′ ε ( x ε ) k X ∗ → , as ε → + . Notice that the above definition reduces to the usual notion of Palais-Smale sequenceswhen I ε ≡ I for every ε > . Let u ∗ ∈ argmin F h, , we denote by K ih ( u ∗ ) the set of criticalpoints in A i of E ih ( u ∗ , · ) that is(2.12) K ih ( u ∗ ) = { w ∈ A i : τ ( u ∗ , w )[ ω ] = 0 , ∀ ω ∈ A i } . Next result shows that any critical point of E h ( u ∗ , · ) in A i can be approximated by auniform Palais-Smale sequence of E ih,ε whose energy converges to the energy of the criticalpoint itself. Theorem 2.3.
Let u ∗ ∈ argmin F h, , w ∈ K ih ( u ∗ ) and z w ∈ argmin Q w ( z ) , where (2.13) Q w ( z ) := Z Ω J (cid:16) E ( z ) + 12 Dw ⊗ Dw (cid:17) d x Then { ( u ∗ + ε z w , w ) } ε> is a uniform Palais-Smale sequence for E ih,ε and lim ε → + E ih,ε ( u ∗ + ε z w , w ) = E ih ( u ∗ , w ) . Proof.
We have to prove the following conditionsa) E ih,ε ( u ∗ + ε z w , w ) ≤ C < + ∞ , ∀ ε ∈ (0 , ,b) ( E ih,ε ) ′ ( u ∗ + ε z w , w ) → strongly in ( H (Ω , R ) × A i ) ∗ ,c) lim ε → + E ih,ε ( u ∗ + ε z w , w ) = E ih ( u ∗ , w ) . We first prove c), which implies a) too. Indeed E ih,ε ( u ∗ + ε z w , w ) = ε − [ F h ( u ∗ + ε z w , εw ) − F ,h ( u ∗ )] == ε − (cid:20) h Z Ω J ( εD w ) dx + h Z Ω J ( E ( u ∗ ) + ε E ( z w ) + ε Dw ⊗ Dw ) d x (cid:21) −− ε − (cid:20) h Z Ω J ( E ( u ∗ )) + ε h Z ∂ Ω f h · z w (cid:21) == ε − (cid:20) h ε Z Ω J ( D w ) d x + h Z Ω J ( E ( u ∗ ) + ε h Z Ω J (cid:18) E ( z w ) + 12 Dw ⊗ Dw (cid:19) d x (cid:21) ++ ε − (cid:20) ε h Z Ω J ′ ( E ( u ∗ )) : (cid:18) E ( z w ) + 12 Dw ⊗ Dw (cid:19) d x − h Z Ω J ( E ( u ∗ )) − ε Z ∂ Ω f h · z w (cid:21) == h Z Ω J ( D w ) d x + ε h Z Ω J (cid:18) E ( z w ) + 12 Dw ⊗ Dw (cid:19) d x ++ h Z Ω J ′ ( E ( u ∗ )) : (cid:18) E ( z w ) + 12 Dw ⊗ Dw (cid:19) d x − h Z ∂ Ω f h · z w == h Z Ω J ( D w ) d x + ε h Z Ω J (cid:18) E ( z w ) + 12 Dw ⊗ Dw (cid:19) d x ++ h Z Ω J ′ ( E ( u ∗ )) : Dw ⊗ Dw d x since, due to minimality of u ∗ , Z Ω J ′ ( E ( u ∗ )) : E ( z w ) d x − Z ∂ Ω f h · z w = 0 . Hence lim ε → E ih,ε ( u ∗ + ε z w , w ) = E ih ( u ∗ , w ) as claimed.Eventually we prove b). By recalling (2.4) and (2.10), we get for every z ∈ H (Ω , R ) and ω ∈ A i ( E ih,ε ) ′ ( u ∗ + ε z w , w )[( z , ω )] = ε − (cid:16) τ (cid:0) u ∗ + ε z w , w (cid:1) [ z ] , τ (cid:0) u ∗ + ε z w , w (cid:1) [ ω ] (cid:17) . Since z w ∈ argmin Q ( z ) , u ∗ ∈ argmin F h, and w ∈ K iw ( u ∗ ) we get: τ ( u ∗ , z ] = 0 ∀ z ∈ H (Ω , R ) , τ ( u ∗ , w )[ ω ] = 0 ∀ ω ∈ A i ,ε − τ ( u ∗ + ε z w , w )[ ω ] = ε Z Ω J ′ (cid:18) E ( z w ) + 12 Dw ⊗ Dw (cid:19) : Dw ⊗ Dω . The above relationships together with (2.12) imply sup k ( z ,ω ) k≤ (cid:12)(cid:12) ( E ih,ε ) ′ ( u ∗ + ε z w , w )[( z , ω )] (cid:12)(cid:12) → as ε → , where k ( z , ω ) k = k z k H + k ω k H , thus proving b ) . (cid:3) Remark 2.4.
Let u ∗ ∈ argmin F h, , w ∈ K ih ( u ∗ ) then(2.14) E ih ( u ∗ , w ) ′ [( , w )] = h Z Ω J ′ ( D w ) · D w d x + h Z Ω J ′ ( E ( u ∗ )) : Dw ⊗ Dw that is E ih ( u ∗ , w ) = 0 and E ih,ε ( u ∗ + ε z w , w ) = ε h min Q w . Remark 2.5.
In Theorem 2.3 we have shown that every critical point for E ih of the kind ( u ∗ , w ) , with u ∗ ∈ argmin F h, and w ∈ K ih ( u ∗ ) , can be approximated (in the strongconvergence of H (Ω , R ) × H (Ω)) by uniform Palais-Smale sequences of E ih,ε . Actuallythe displacement pair sequence can be chosen explicitly of the kind ( u ∗ + ε z w , w ) , say withfixed out-of-plane component and in-plane displacement approximated by an infinitesimalcorrection tuned by the out-of-plane component. Nevertheless we cannot expect that everyuniform Palais-Smale sequence of E ih,ε is equibounded in H (Ω , R ) × A i , as we are goingto show in the next Counterexample. Counterexample 2.6. ( a uniform Palais-Smale sequence lacking compactness ) If Ω = (0 , a ) × (0 , , Γ ≡ ∂ Ω and f h = γ e ( (0 ,a ) ×{ } ) − (0 ,a ) ×{ } ) , where γ is a suitableconstant to be chosen later, then the unboundedness may develop.So by Theorem 1.8 (clamped plate), ∀ h > , ∀ ε > there exists ( u ε , w ε ) ∈ argmin E h,ε .Hence ( u ε , w ε ) is a uniform Palais-Smale sequence for E h,ε , moreover we show below thatsuch a sequence must lack weak compactness in H (Ω , R ) × H (Ω) for big γ . Indeed, ifcompactness were true, we would obtain (up to subsequences) that ( u ε , w ε ) ⇀ ( u , w ) ∈ argmin E h , due to Proposition 2.1. Eventually we show that inf E h = −∞ , thus obtaininga contradiction.Actually, due to Euler equations(2.15) Z Ω J ′ (cid:0) E ( u ) (cid:1) : E ( v ) = Z ∂ Ω f h · v = − γ Z Ω v , ∀ v ∈ H (Ω , R ) , so, for every u ∈ argmin F h, , J ′ ( E ( u )) = − γ e ⊗ e , u = 2 γνE ν ν ( x e + x e ) + r , r ∈ R , and by (2.5)(2.16) E h ( u , w ) = h Z Ω J ( D w ) − hγ Z Ω | w , | d x , if u ∈ argmin F h, ( · , , w ∈ A , + ∞ otherwise in H (Ω) × H (Ω) . Hence, if u ∈ argmin F h, , w ∈ A , we get E h ( u , w ) ≤ C ν E h Z Ω | D w | d x − h γ Z Ω | w , | d x . Set w ( x , x ) = α ( x ) β ( x ) , with α ∈ H (0 , a ) and β ∈ H (0 , . Then w ∈ H (Ω) and(2.17) E h ( u , w ) ≤ ( A C + A C + A ) C ν Eh Z | β ′′ | dx − A hγ Z | β ′ | dx , where A = Z | α ′′ | dx , A = 2 Z | α ′ | dx , A = Z α dx and C , C are the best constants such that Z β dx ≤ C Z | β ′′ | dx , Z | β ′ | dx ≤ C Z | β ′′ | dx ∀ β ∈ H (0 , If ξ ∈ H (0 , is the eigenfunction fulfilling the equality R | ξ ′ | dx = C R | ξ ′′ | dx and γ >
16 ( A C + A C + A ) C ν E h / ( A C ) . Setting β n := nξ ∈ H (0 , and w = αβ n , the right-hand side of (2.17) goes to −∞ as n → ∞ . In the previous counterexample we have shown that some uniform Palais-Smale sequencemay be not converging to any critical point, while in the next examples we show howTheorem 2.3 can be used to detect buckled configurations of the plate (associated tocritical points for
FvK ) by means of uniform Palais-Smale sequences for the approximatingfunctionals.
Example 2.7. ( buckling of a rectangular plate under compressive load ) Set
Ω = (0 , a ) × (0 , , f h = γ e ( (0 ,a ) ×{ } ) − (0 ,a ) ×{ } ) and Γ = Σ + ∪ Σ − , with Σ + = [0 , × { } , Σ − = [0 , × { } .Now Γ = ∂ Ω : by arguing as the in previous Counterexample we find noncompact uniformPalais-Smale sequences together with energy of admissible configurations unbounded frombelow.In the present case we push forward the analysis: as before we find that if u ∈ argmin F h, and w ∈ A i , i=0,1,2, then J ′ ( E ( u )) = − γ e ⊗ e , so that E ih ( u , w ) = h Z Ω J ( D w ) d x − hγ Z Ω | w , | d x if u ∈ argmin F ih, ( · , , w ∈ A i . We look for critical points in the form w = w ( x ) under the following conditions: w (0) = w (1) = w ′ (0) = w ′ (1) = 0 , if i = 0 ; w (0) = w (1) = 0 , if i = 1 ; w (0) ′′ = w ′′ (1) = w ′′′ (0) = w ′′′ (1) = 0 , if i = 2 .Since J ( e ⊗ e ) = E − ν ) , we have E ih ( u , w ) = Eh − ν ) Z | w ′′ ( x ) | dx − hγ a Z | w ′ ( x ) | dx whose non-trivial critical points can be easily computed, via the ODE w ′′′′ + 12 γa (1 − ν ) Eh w ′′ = 0 . Theorem 2.3 allows to recover Palais-Smale sequences for E ih,ε , i = 0 , , .In the clamped case ( i = 0 ) the nontrivial buckled solutions occur for discrete choices of h : h n = 12 n π r γ a (1 − ν ) E , w n ( x ) = 1 + sin r γ a (1 − ν ) E h ( x − π/ ! , n ∈ N ; else, for any other choice of h , w ≡ .The associated Palais-Smale sequence is (cid:0) γνE ν ν ( x e + x e ) + ε z w n ( x , x ) , w n ( x ) (cid:1) ,where z w n ( x , x ) = (cid:16) , / R x | w ′ n ( t ) | dt (cid:17) and w n is given above. Example 2.8. ( buckling of a rectangular plate under shear load ). Set
Ω = ( − , × ( − , , i = 0 , and Γ = Σ , ± ∪ Σ , ± , where: Σ , + = [ − , × { } , Σ , − = [0 , × {− } , Σ , + = { } × [ − , , Σ , − = {− } × [ − , .Assume f h = γ τ (cid:0) S , ± − S , ± (cid:1) , where S , ± = Σ , ± , S , ± = [ − , × {± } , γ > , τ is thecounterclockwise oriented tangent unit vector to ∂ Ω = S , ± ∪ S , ± .Since u ∈ argmin F h, , by exploiting Euler-Lagrange equations as before, we obtain J ′ ( E ( u )) = γ ( e ⊗ e + e ⊗ e ) and by (2.5) E h ( u , w ) = h Z Ω J ( D w ) dx + hγ Z Ω w , w , d x . We look for critical points in the form(2.18) w = ψ ( x − x ) if ( x , x ) ∈ Ω , | x − x | ≤ else in Ω , and satisfying ψ ( ±
1) = ψ ′ ( ±
1) = 0 .By J ( e ⊗ e + e ⊗ e − e ⊗ e − e ⊗ e ) = 2 E − ν we obtain E h ( u , w ) = h E − ν ) Z − | ψ ′′ ( t ) | dt − hγ Z − | ψ ′ ( t ) | dt whose nontrivial critical points can be easily computed, via the ODE ψ ′′′′ + 6 γ (1 − ν ) Eh ψ ′′ = 0 , ψ ( ±
1) = ψ ′ ( ±
1) = 0 . Therefore even now the nontrivial buckled solutions occur for (different) discrete choicesof h : w = w n ( x , x ) = ψ n ( x − x ) := 1 + sin r γ a (1 − ν ) E h n ( x − x + 1 / ! if h n = 1 n π r γ a (1 − ν ) E , with n ∈ N ; else, we have the flat solution w ≡ for any other choice of h .The associated Palais-Smale sequence is (cid:0) u ( x , x ) + ε z w n ( x , x ) , w n ( x , x ) (cid:1) , where u ( x , x ) = γ νE ( x , x ) , z w n ( x , x ) = (cid:16) − (1 / R x − x − | w ′ n ( t ) | dt , (1 / R x − x − | w ′ n ( t ) | dt (cid:17) . Remark 2.9.
In Examples 2.7, 2.8, when nontrivial solutions exist the period of theoscillations has order h . By scaling loads, that is by taking f h = h α f , we get J ′ ( E ( u )) = − h α γ ( e ⊗ e ) and J ′ ( E ( u )) = h α γ ( e ⊗ e + e ⊗ e ) respectively, while related limitfunctionals become respectively E ih ( u , w ) = Eh − ν ) Z | w ′′ ( x ) | dx − h α +1 γ a Z | w ′ ( x ) | dx , i = 0 , , , E h ( u , w ) = h E − ν ) Z − | w ′′ ( t ) | dt − h α +1 γ Z − | w ′ ( t ) | dt, whose nontrivial critical points obviously exhibit oscillation period of order h − α/ .Computations in Remark 2.9 proves useful in the next Section when studying asymptoticsof the problem as the thickness tends to + .3. Scaling Föppl-von Kármán energy
Here we focus on the asymptotic analysis of the mechanical problems for
Fvk plate as h → + . To highlight properties of the limit solution we examine the behavior of suitablyscaled energy: all along this Section we assume that there is no transverse load, say g h ≡ ,while we refer to a parameter α characterizing different asymptotic regimes of in-plane load f h , say(3.1) f h = h α f where α ≥ and f ∈ L ( ∂ Ω , R ) . The next result and the subsequent counterexample show how the choice of α may influencethe asymptotic behavior of functionals F h when h → + . Theorem 3.1.
Let Ω ⊂ R be a bounded connected Lipschitz open set, α ≥ and i = 0 , , .If i = 0 (clamped plate) assume (1.29) and Γ = ∂ Ω (as in Theorem 1.8) .If i = 1 (simply supported plate) assume (1.29) , Ω strictly convex, Γ = ∂ Ω (as in Theorem1.6).If i = 2 (free plate) assume (1.12) and (1.13) (as in Theorem 1.1).Set (3.2) F i,α ( v , ζ ) = F i ( v , ζ ) if α = 2 F i ( v , ζ ) + χ { D ζ ≡ } ( ζ ) if α > , where χ { D ζ ≡ } ( ζ ) = 0 if D ζ ≡ , = + ∞ else.Fix i ∈ { , , } and a sequence ( u h , w h ) in argmin F ih .Then there exists ( v , ζ ) ∈ argmin F i,α such that, up to subsequences, (3.3) ( h − α u h , h − α/ w h ) → ( v , ζ ) weakly in H (Ω , R ) × H (Ω) , as h → + . Moreover (3.4) h − α − F ih ( u h , w h ) → F i,α ( v , ζ ) , as h → + . Proof.
The case α = 2 is trivial since ( u h , w h ) ∈ argmin F ih if and only if ( h − u h , h − w h ) ∈ argmin F i for every h .If α > , i = 0 , and ( u h , w h ) ∈ argmin F ih , set v h := h − α u h , ζ h := h − α/ w h , λ h = k E ( v h k L and assume by contradiction λ h → + ∞ . Then by taking into account minimalityof ( u h , w h ) , (1.7), (1.29) and setting ϕ h = λ − / h ζ h , z h = λ − h v h we get(3.5) c ν h − α E Z Ω | D ϕ h | + λ h c ν E Z Ω | D ( z h , ϕ h ) | ≤ Z ∂ Ω f · z h ≤ C. Hence | D ϕ h | → in L (Ω , Sym , ( R )) and by taking into account that ϕ h = 0 on ∂ Ω we get ϕ h → in H (Ω) ; therefore E ( z h ) → O in L (Ω , R ) , a contradiction since k E ( z h ) k L = 1 .Then λ h is bounded from above and by taking into account minimality of ( u h , w h ) , (1.7),(1.29) we get(3.6) c ν h − α E Z Ω | D ζ h | + c ν E Z Ω | D ( v h , ζ h ) | ≤ Z ∂ Ω f · v h ≤ k f k λ h ≤ C which entails D ζ h → in L (Ω) and equiboundedness of Dζ h in L (Ω , R ) .When i = 2 we take again λ h = k E ( v h ) k L and assume by contradiction λ h → + ∞ .Then estimate (3.5) continues to hold and as before | D ϕ h | → in L (Ω) which entails ϕ h − − R Ω ϕ h → in L (Ω) , Dϕ h → c in L and E ( z h ) → − c ⊗ c strongly in L (Ω , Sym , ( R )) for a suitable c ∈ R . Therefore (1.13), (3.5) yield(3.7) ≤ lim h → + Z ∂ Ω f · z h = lim h → + f Z ∂ Ω n · z h = lim h → + f Z Ω div z h = − f | Ω || c | that is c = so E ( z h ) → O in L (Ω , Sym , ( R )) as in the previous cases, again a contra-diction. Thus equiboundedness holds in this case too. Since, for < h ≤ , the w.l.s.c.functionals F i,α fulfil F i,α ≤ h − α − F ih , the proof can be completed by a standard argumentin Γ convergence. (cid:3) Remark 3.2.
It is worth noticing that when D w ≡ then(3.8) F ( v , w ) = F ( v , , if i = 0 , , (3.9) F ( v , w ) = F ( v , ξ · x ) = Z Ω J ( E ( v ) + 12 ξ ⊗ ξ ) − Z ∂ Ω f h · v for w = ξ · x , if i = 2 . Theorem 3.1 is optimal in the sense that if α < we cannot expect neither that h − α − min A i F h are bounded from below nor that minimizers are equibounded in H (Ω , R ) × H , (Ω) whenwe let h → + . This phenomenon may take place even if Ω is a rectangle as shown bythe next Counterexample, where we consider a plate with the same geometry and load ofCounterexample 2.6 , nevertheless here we push further the analysis of this case. Counterexample 3.3.
Let a > EC ν , α ∈ [0 , , f h = h α f with(3.10) Ω = (0 , a ) × (0 , , Γ = ∂ Ω , g h ≡ , f = (cid:0) { y =0 } − { y =1 } ) e . Then for any sequence ( u h , w h ) ∈ arg min F h (such sequences do exist due to Theorem1.8), the scaled sequence ( h − α u h , h − α/ w h ) is not equibounded in H (Ω , R ) × H , (Ω) .Moreover, inf h − α − F h → −∞ as h → + .Indeed we can set: v h := h − α u h , ζ h := h − α/ w h , and(3.11) W h ( v h , ζ h ) := h − − α F h ( u h , w h ) = h − α Z Ω J ( D ζ h )+ Z Ω J ( D ( v h , ζ h )) − Z ∂ Ω f · v h , (3.12) I + ( v , ζ ) := inf ( lim sup h → + W h ( v h , ζ h ) : v h w − H ⇀ v , ζ h w − H , ⇀ ζ ) , (3.13) I − ( v , ζ ) := inf (cid:26) lim inf h → + W h ( v h , ζ h ) : v h w − H ⇀ v , ζ h w − H , ⇀ ζ (cid:27) , (3.14) J ( B , η ) = E ν ) (cid:12)(cid:12) B + B T + η ⊗ η (cid:12)(cid:12) + Eν − ν ) (cid:12)(cid:12) Tr (cid:0) B + B T + η ⊗ η (cid:1)(cid:12)(cid:12) . Then by arguing as in Lemma 4.1 of [14] we get(3.15) I + ( v , ζ ) ≤ Λ( v , ζ ) := Z Ω J ( D ( v , ζ )) dx − Z ∂ Ω f · v d H . Then by denoting with Q J the quasiconvex envelope of J , since I + is sequentially lowersemicontinuous in w − H × w − H , , we obtain(3.16) I + ( v , ζ ) ≤ Z Ω Q J ( D v , Dζ ) dx − Z ∂ Ω f · v d x . On the other hand for every v h w − H ⇀ v , ζ h w − H , ⇀ ζ we get lim inf h → + h − − α F h ( h α v h , h α/ ζ h ) ≥ Z Ω Q J ( D v , Dζ ) − Z ∂ Ω f · v that is I − ( v , ζ ) ≥ Z Ω Q J ( D v , Dζ ) − Z ∂ Ω f · v . By(3.17) I ( v , ζ ) := Z Ω Q J ( D v , Dζ ) − Z ∂ Ω f · v ≥ I + ( v , ζ ) ≥ I − ( u , w ) ≥ I ( v , ζ ) we get(3.18) Γ lim h → + W h = I . Therefore, if ( h − α u ∗ h , h − α/ w ∗ h ) were equibounded in H (Ω , R ) × H , (Ω) then h − − α F h ( u ∗ h , w ∗ h ) → min I = inf Λ since Λ is the relaxed functional of I , and we will show that this leads to a contradiction.Indeed, we choose(3.19) ζ n ( x, y ) = 1 √ n ϕ ( ny ) ψ n ( x ) , v n ( x, y ) = (0 , − n y ) , with(3.20) ϕ : R → R , -periodic , ϕ ( y ) = 12 (1 − | − y | ) ∀ y ∈ (0 , (3.21) ψ n ( x ) = nx { [0 , /n ] } + { [1 /n,a − /n ] } − n ( x − a ) { [ a − /n,a ] } . We get E ( v n ) = " − n , D ( v n , ζ n ) = n (cid:0) ψ ′ n ( x ) (cid:1) (cid:0) ϕ ( ny ) (cid:1) ψ n ( x ) ψ ′ n ( x ) ϕ ( ny ) ϕ ′ ( ny )12 ψ n ( x ) ψ ′ n ( x ) ϕ ( ny ) ϕ ′ ( ny ) n (cid:0) ψ n ( x ) | ϕ ′ ( ny ) | − (cid:1) and by taking into account (1.6), (1.7) and that | ϕ | ≤ , | ϕ ′ | = 1 , | ψ | ≤ , | ψ ′ n | ≤ n, spt ψ ′ n ⊂ [0 , /n ] ∪ [ a − /n, a ] , | ψ n | = 1 on [1 /n, a − /n ] , a > EC ν , Λ( v n , ζ n ) = Z a Z J ( D ( v n , ζ n ) ) dx dy − Z ∂ Ω f · v n dx dy ≤≤ Z a Z EC ν (cid:16) n − | ψ ′ n ( x ) | | ϕ ( ny ) | +2 | ψ n ( x ) | | ψ ′ n ( x ) | | ϕ ( ny ) | | ϕ ′ ( ny ) | + n (cid:0) ψ n ( x ) | ϕ ′ ( ny ) | − (cid:1) (cid:17) − na ≤ Z a Z EC ν (cid:16) n [0 , /n ] ∪ [ a − /n,a ] + n (cid:0) ψ n ( x ) − (cid:1) (cid:17) − na ≤ nEC ν − na → −∞ . leads to a contradiction.So ( h − α u ∗ h , h − α/ w ∗ h ) are not equibounded in H (Ω , R ) × H , (Ω) and the first claim follows.Eventually we prove the second claim. By (3.15) there exists ( v n,h , ζ n,h ) → ( v n , ζ n ) weaklyin H (Ω , R ) × H (Ω) such that lim sup W h ( v n,h , ζ n,h ) ≤ I ( v n , ζ n ) ≤ − Kn for suitable K > , hence by using a diagonal argument we achieve the claim. Remark 3.4. If a > EC ν , α ∈ [0 , , f h = h α f with(3.22) Ω = (0 , a ) × (0 , , g h ≡ , f = (cid:0) { y =0 } − { y =1 } ) e , Γ = ∂ Ω , Then h − − α inf F ih → −∞ as h → + holds true also for i = 1 , .Indeed, though existence of minimizers of F ih , ( i = 1 , may fail, nevertheless inf F ih ≤ inf F h for i = 1 , ; hence the claim follows by previous Counterexample. Prestressed plates: oscillating versus flat equilibria.
Counterexample 3.3 and Remark 3.4 show that the Föppl Von Karman functional might notbe suitable for studying equilibria of plates when thickness h → + , at least in presenceof in-plane loads scaling as h α , when α ∈ [0 , and h is the scale factor for the platethickness.To circumvent this difficulty, as in the case of many practical engineering applications,we assume that our plate-like structure is initially prestressed and undergoes a transversedisplacement about the prestressed state.Precisely, in this Section we fix g h ≡ , f ∈ L ( ∂ Ω , R ) , α ∈ [0 , and we assume thatthe prestressed state is caused by the (scaled) force field f h = h α f and is given by every u ∗ ∈ H (Ω , R ) , u ∗ = h α v ∗ where v ∗ is a minimizer of the functional(4.1) F ( v ) := Z Ω J ( E ( v )) − Z ∂ Ω f · v The transverse displacement w is chosen such that the pair ( u ∗ , w ) minimizes the functional G h over H (Ω , R ) × A i , defined by G h ( u , w ) = (cid:26) F h ( u , w ) if u = u ∗ and w ∈ A i , + ∞ else . Moreover we have G h ( u , w ) = e G h ( v , ζ ) when setting v := h − α u , ζ := h − α/ w and e G h ( v , ζ ) = h α F bh ( ζ ) + h α +1 Z Ω J ( D ( v , ζ )) − h α +1 Z ∂ Ω f · v , if v ∈ argmin F , ζ ∈ A i , + ∞ else in H (Ω) × A i . We aim to capture the nature of the transverse minimizer through a detailed study of theasymptotic behavior of minimizers of e G h as h → + . A first hint in this perspective is thenext result. Theorem 4.1.
For every v ∈ arg min F , let I ∗∗ v ( x , · ) be the convex envelope of I v ( x , . ) where I v ( x , ξ ) := J (cid:18) E ( v )( x ) + 12 ξ ⊗ ξ (cid:19) , and (4.2) G ∗∗ ( v , ζ ) := Z Ω I ∗∗ v ( x , Dζ ) d x − Z ∂ Ω f · v d H ∀ ζ ∈ H , (Ω) . Then, for every α ∈ [0 , , (4.3) h − α − min A i e G h → min {G ∗∗ ( v , ζ ) : ζ ∈ H , (Ω) , ζ = 0 on Γ } if i = 0 , {G ∗∗ ( v , ζ ) : ζ ∈ H , (Ω) } if i = 2 . Moreover if ( v , ζ h ) ∈ arg min A i e G h then ζ h → ζ weakly in H , (Ω) , up to subsequences, with ( v , ζ ) ∈ arg min G ∗∗ .Proof. The claim is a straightforward consequence of techniques developed in Lemma 4.1of [14] and standard relaxation of integral functionals. (cid:3)
In order to characterize equilibrium configurations of e G h , additional information about min-imizers of functional G ∗∗ are needed: actually a careful use of Theorem 4.1 allows to showexplicit examples capturing the qualitative behavior of minimizers and their dependanceon the thickness h .To this aim, if A ∈ Sym , ( R ) we denote its ordered eigenvalues by λ ( A ) ≤ λ ( A ) andby v ( A ) , v ( A ) their corresponding normalized eigenvectors, which afterwards will be de-noted shortly with λ , λ , v , v whenever there is no risk of confusion.For every ν = 1 , ξ ∈ R and A ∈ Sym , ( R ) we set(4.4) g A ( ξ ) = | A + ξ ⊗ ξ | + ν (1 − ν ) (cid:0) Tr A + | ξ | (cid:1) . Lemma 4.2. If ν ∈ ( − , / , then (4.5) min ξ ∈ R g A ( ξ ) = g A ( ) if νλ + λ ≥ ν )( λ ( A )) if νλ + λ < . Proof.
It is worth noticing that minimum in (4.5) is achieved since g A ∈ C ( R ) and g A ( ξ ) → + ∞ as | ξ | → + ∞ . Let M ∈ O (2) be such that M T AM = diag ( λ , λ ) . Then it is readilyseen that by setting x := ξ · v , y := ξ · v we have ˜ g A ( x, y ) := g A ( ξ ) = ( x + λ ) + ( y + λ ) + 2 x y + ν − ν (cid:0) λ + λ + x + y ) (cid:1) and an easy computation shows that if νλ + λ ≥ then minimum is attained at ( x, y ) =(0 , . Else, if νλ + λ < then either νλ + λ ≥ or νλ + λ ≤ νλ + λ < .In the first case D ˜ g A ( x, y ) = (0 , if and only if ( x, y ) ∈ { ( ±√− νλ − λ , , (0 , } and ˜ g A ( x, y ) = (1 + ν ) λ or g A ( x, y ) = g A (0 , > (1 + ν ) λ ; in the latter one D ˜ g A ( x, y ) = (0 , also at ( x ∗ , ± y ∗ ) = (0 , ±√− νλ − λ ) with ˜ g A ( x ∗ , ± y ∗ ) = (1 + ν ) λ . Hence min ξ ∈ R g A ( ξ ) = (1 + ν ) λ if νλ + λ < ≤ νλ + λ and min ξ ∈ R g A ( ξ ) = (1 + ν ) min { λ , λ } if νλ + λ ≤ νλ + λ < . In the latter case if ν ∈ ( − , then λ ≤ λ ≤ − νλ , hence λ ≤ λ ≤ and | λ | ≥ | λ | . If ν ∈ [0 , / then λ < and either λ > | λ | > or λ ≤ λ ≤ . In the first case we get necessarily ν > and | λ | > ν − (1 − ν ) λ > λ , acontradiction. Therefore | λ | ≤ | λ | and min ξ ∈ R g A ( ξ ) = (1 + ν ) λ whenever νλ + λ < thus proving the thesis. (cid:3) Lemma 4.2 proves quite useful in the perspective of the next Proposition and the subsequentExamples, since the two alternatives in the right-hand side of (4.5) correspond respectivelyto locally flat or oscillating equilibrium configurations.
Proposition 4.3. If v ∗ ∈ argmin F and the ordered eigenvalues λ ≤ λ of E ( v ∗ ) fulfil νλ + λ ≥ in the whole set Ω , then (4.6) e G h ( v ∗ , ζ ) ≥ e G h ( v ∗ , . If in addition νλ + λ > in a set of positive measure, then the inequality in (4.6) is strictfor every ζ .Proof. Due to (4.5) in Lemma 4.2: νλ + λ ≥ entails g E ( u ∗ ) ( ξ ) ≥ g E ( u ∗ ) ( ) , moreover νλ + λ > entails g E ( u ∗ ) ( ξ ) > g E ( u ∗ ) ( ) . Hence J (cid:0) D ( v ∗ , ζ ) (cid:1) = E ν ) g E ( v ∗ ) ( Dw ) ≥ J (cid:0) E ( v ∗ ) (cid:1) and, for ζ ∈ A i , e G h ( v ∗ , ζ ) = h α F bh ( ζ ) + h α +1 Z Ω J (cid:0) D ( v ∗ , ζ ) (cid:1) − h α +1 Z ∂ Ω f · v ∗ ≥≥ h α F bh ( ζ ) + h α +1 Z Ω J (cid:0) E ( v ∗ ) (cid:1) − h α +1 Z ∂ Ω f · v ∗ ≥ e G h ( v ∗ , . Moreover the first inequality in the last computation is strict whenever νλ + λ > in aset of positive measure. (cid:3) Remark 4.4.
Notice that s := E − ν ( νλ + λ ) is the smallest eigenvalue of the stresstensor T ( v ) = J ′ (cid:0) E ( v ) (cid:1) . Therefore Proposition 4.3 shows that, if the eigenvalues of thestress tensor are both strictly positive almost everywhere, then we can expect only one flatminimizer ( ζ ≡ ). On the other hand, the possible occurrence of oscillating configurationsrequires the presence of a compressive state on a region of positive measure: that is to saythe stress tensor must have at least one negative eigenvalue on set of positive measure.We show some examples clarifying how the asymptotic behavior of functionals e G h providesuseful information about minimizers when Ω is an annular set. Set < R < R , p , p ∈ R , Ω := B R \ B R , and consider uniform in-plane normaltraction/compression at each component of the boundary. f = − p x R {| x | = R } + p x R {| x | = R } . Therefore v ∈ arg min F , entails(4.7) v ( x ) = ( a + b | x | − ) x , and exploiting polar coordinates x = ( r cos θ, r sin θ ) we obtain E ( v ) = a − br cos 2 θ − br sin 2 θ − br sin 2 θ a + br cos 2 θ . By using Neumann boundary condition J ′ ( E ( v )) n = f on ∂ Ω , we get :(4.8) p i = E (1 + ν ) − ( a (1 + ν )(1 − ν ) − − bR − i ) , i = 1 , that is a = (1 − ν )( p R − p R ) E ( R − R ) ; b = (1 + ν )( p − p ) R R E ( R − R ) . It is worth noticing that a − br − , a + br − are the eigenvalues of E ( v ) and (cos θ, sin θ ) , ( − sin θ, cos θ ) the corresponding normalized eigenvectors ∀ r ∈ [ R , R ] ; order may change according to sign( b ) .We examine several different cases which may occur. Example 4.5.
Radially oscillating minimizers.
Set
Γ = ∂ Ω , ν ∈ ( − , / , i = 0 andeither p ≤ p < or p ≤ p < . In the first case we get b ≥ in the second one b ≤ .However in both cases νλ + λ < in the whole annular set.Set also v ( x ) = ( a + b | x | − ) x ∈ arg min F , .Choose σ h → + , β h → + ∞ , ψ h : R → R ( R − R ) -periodic such that(4.9) ψ h ( t ) = max { , min { t − R − σ h , R − σ h − t }} and set ψ ∗ h := ψ h ∗ ρ h being ρ h mollifiers such that spt ρ h ⊂ [ − σ h , σ h ] . Then by denoting thefloor of a real number (maximum integer not exceeding the number) with ⌊·⌋ and setting r = | x | , ζ h ( r ) = ⌊ β h ⌋ − p − ν ) br − − a ( ν + 1) ψ ∗ h ( R + ( r − R ) ⌊ β h ⌋ ) if p ≤ p < , ⌊ β h ⌋ − p ν − br − − a ( ν + 1) ψ ∗ h ( R + ( r − R ) ⌊ β h ⌋ ) if p ≤ p < ,ζ ′ h := ∂ζ /∂r , Dζ h = ( ζ h, , ζ h, ) = ( x /r, x /r ) ζ ′ h and M ( θ ) = cos θ − sin θ sin θ cos θ. S ( θ ) = cos θ sin θ cos θ sin θ cos θ sin θ = ( ζ ′ h ) − Dζ h ⊗ Dζ h . So M T S M = e ⊗ e and there exists Ω h ⊂ Ω with | Ω h | ∼ σ h such that, | ( ψ ∗ h ) ′ | = 1 on Ω \ Ω h . Then referring to (4.4) and (4.7), for every x ∈ Ω \ Ω h we have g E ( v ) ( Dζ h ) = | E ( v ) + Dζ h ⊗ Dζ h | + ν − ν | v + | Dζ h | | = (cid:12)(cid:12) M T E ( v ) M + M T Dζ h ⊗ Dζ h M (cid:12)(cid:12) + ν − ν | a + | Dζ h | | = (cid:12)(cid:12) a − br − ) e ⊗ e + 2( a + br − ) e ⊗ e + | ζ ′ h | M T S M (cid:12)(cid:12) + ν − ν | a + | ζ ′ h | | =(2 a − br − + | ζ ′ h | ) + 4( a + br − ) + ν − ν | a + | ζ ′ h | | . If p ≤ p < , we have b ≥ , | ζ ′ h | = 2(1 − ν ) br − − a ( ν + 1) + O ( ⌊ β h ⌋ − ) on Ω \ Ω h ,hence g E ( v ) ( Dζ h ) = 4(1 + ν )( a + br − ) + O ( ⌊ β h ⌋ − ) , Z Ω I v ( x , Dζ h ) d x = Z Ω \ Ω h I v ( x , Dζ h ) d x + Z Ω h I v ( x , Dζ h ) d x == E − ν ) Z Ω \ Ω h { ( a + b | x | − ) + O ( β − h ) } d x + O ( σ h ) → E − ν ) Z Ω ( a + b | x | − ) dx Analogously, if p ≤ p < , then b ≤ and | ζ ′ h | = 2( ν − br − − a ( ν + 1) + O ( ⌊ β h ⌋ − ) on Ω \ Ω h , hence g E ( v ) ( Dζ h ) = 4(1 + ν )( a − br − ) + O ( ⌊ β h ⌋ − ) , Z Ω I v ( x , Dζ h ) d x → E − ν ) Z Ω ( a − b | x | − ) d x . By Lemma 4.2 we know that min ξ ∈ R I v ( x, ξ ) = E − ν ) ( a + b | x | − ) if p ≤ p < ,E − ν ) ( a − b | x | − ) if p ≤ p < , therefore in both cases we have proved that Z Ω I v ( x , Dζ h ) dx → min {G ∗∗ ( v , ζ ) : ζ ∈ H , (Ω) , ζ = 0 in ∂ Ω } . Moreover h − α − e G ( v , ζ h ) = h − α − F bh ( ζ h ) + Z Ω I v ( x, Dζ h ) d x − Z ∂ Ω f · v d H ,h − α − F bh ( ζ h ) ∼ h − α β h σ − h . Therefore by Theorem 4.1 for every choice of β h , σ h satisfying the conditions detailedbefore, ( v , ζ h ) can be viewed as an asymptotically minimizing sequence of e G h whose out-of-plane component exhibits periodic oscillations (period: β − h ; asymptotic amplitude: p − ν ) br − − a ( ν + 1) if p ≤ p < and p ν − br − − a ( ν + 1) if p ≤ p < )in the radial direction in the whole annular set. The optimal choice of β h can be deter-mined heuristically as follows: previous estimates show that h − α − e G ( v , ζ h ) − min G ∗∗ = R h where R h ∼ h − α β h σ − h + β − h + σ h . So, approximatively, we have to minimize the lastterm. A direct calculation shows that the best choice corresponds to β h − ∼ h / − α/ , σ h ∼ h / − α ) . Example 4.6.
Flat minimizer.
Let
Γ = ∂ Ω , ν ∈ [0 , / , i = 0 or i = 1 , p ≥ ,so that R a ≥ (1 − ν ) b and by Lemma 4.2 we get min {G ∗∗ ( v , ζ ) : ζ ∈ H , (Ω) , ζ = 0 in ∂ Ω } = Z Ω I v ( x , d x . Obviously the minimum is attained at ζ ≡ that is we have a flat minimizer. Remark 4.7.
Let
Γ = ∂ Ω , ν ∈ ( − , / , i=0 , p < ≤ p . Hence a > , b > , νλ + λ = a − br − + ν ( a + br − ) ≥ in the annular set A = { R := p (1 − ν )(1 + ν ) − ba − ≤ r ≤ R } and < in the annular set A = { R ≤ r < R } . Then by the same computationsperformed in previous examples we can build minimizers which are flat in A and oscillatingin A . Example 4.8.
Tangentially oscillating minimizers.
Let
Γ = ∂B R , ν ∈ ( − , / , i = 1 and choose p > , p > such that p R = p R . If v ∈ arg min F , we find again v ( x ) = ( a + b | x | − ) x with(4.10) a = 0 , b = − (1 + ν ) E − p R < . Hence λ = br − < < − br − = λ are the eigenvalues of E ( v ) and v = ( − sin θ, cos θ ) , v =(cos θ, sin θ ) the corresponding normalized eigenvectors.Choose σ h → + , β h → + ∞ , φ h : R → R , π -periodic defined by(4.11) φ h ( t ) = max { , min { t − σ h , π − σ h − t }} and set φ ∗ h := φ h ∗ ρ h being ρ h mollifiers such that spt ρ h ⊂ [ − σ h , σ h ] . Let ζ h ( r, θ ) = p − b (1 − ν ) ⌊ β h ⌋ − φ ∗ h ( ⌊ β h ⌋ θ ) (cid:0) δ − h ( r − R ) [ R ,R + δ h ] ( r ) + [ R + δ h ,R ] ( r ) (cid:1) with δ h → + , β − h δ − h → . Then there exists Ω h ⊂ Ω with | Ω h | ∼ σ h such that for every x ∈ Ω \ Ω h we have | ( φ ∗ h ) ′ | = 1 on Ω \ Ω h . Therefore referring to (4.4) and (4.7) and bysetting R ∗ ( θ ) = − sin θ cos θ cos θ sin θ we get Z Ω \ Ω h (cid:18) | E ( v ) + Dζ h ⊗ Dζ h | + ν − ν | Dζ h | (cid:19) d x == Z Ω \ Ω h (cid:18)(cid:12)(cid:12) R T ∗ E ( v ) R ∗ + R T ∗ Dζ h ⊗ Dζ h R ∗ (cid:12)(cid:12) + ν − ν | Dζ h | (cid:19) d x == Z Ω \ Ω h ν ) b | x | − d x + O ( ⌊ β h ⌋ − δ − h ) + O ( σ h ) + O ( δ h ) . By using now Lemma 4.2 and by arguing as in Example 4.5 we get Z Ω I v ( x , Dζ h ) dx − Z ∂ Ω f · v d H → min {G ∗∗ ( v , ζ ) : ζ ∈ H , (Ω) , ζ = 0 in ∂ Ω } .h − α − e G ( v , ζ h ) → min G ∗∗ = Eb ν ) Z Ω | x | − d x − Z ∂ Ω f · v d H . Moreover, since h − α − F bh ( ζ h ) ∼ h − α β h σ − h , we get h − α − e G ( v , ζ h ) = h − α − F bh ( ζ h ) + R Ω I v ( x, Dζ h ) d x − R ∂ Ω f · v d H == h − α β h σ − h + O ( ⌊ β h ⌋ − δ − h ) + O ( σ h ) + O ( δ h ) . Hence, here the optimal choice is β h − ∼ h − α/ , δ h ∼ β − / h , σ h ∼ h − α/ β / h . Remark 4.9.
Thanks to Lemma 4.2 and Proposition 4.3, Examples 4.5, 4.6, 4.8 constitutea paradigm for the construction of oscillating versus flat approximated minimizers.Moreover we sketch another technique to devise new ones, by this procedure: first take aboundary force field, construct the corresponding prestressed state (in D there are a lot ofsignificant classical examples, see for instance those of Examples 2.7, 2.8) and look at theeigenvalues of the strain matrix: it is not difficult to obtain examples according to either νλ + λ ≥ or νλ + λ < in the whole plate.In the first case through Lemma 4.2 and Proposition 4.3 we argue that there is only a flatminimizer, in the second one a careful use of Lemma 4.2 on the pattern of Examples 4.5,4.8 allows an easy construction. References [1] G. Anzellotti, S. Baldo, D. Percivale,
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