A C 0 Interior Penalty Method for the von Kármán Equations

Transcription

1 Louisiana State University LSU Digital Commons LSU Doctoral Dissertations Graduate School 2011 A C 0 Interior Penalty Method for the von Kármán Equations Armin Karl Reiser Louisiana State University and Agricultural and Mechanical College, areise1@lsu.edu Follow this and additional works at: Part of the Applied Mathematics Commons Recommended Citation Reiser, Armin Karl, "A C 0 Interior Penalty Method for the von Kármán Equations" (2011). LSU Doctoral Dissertations This Dissertation is brought to you for free and open access by the Graduate School at LSU Digital Commons. It has been accepted for inclusion in LSU Doctoral Dissertations by an authorized graduate school editor of LSU Digital Commons. For more information, please contactgradetd@lsu.edu.

2 A C 0 INTERIOR PENALTY METHOD FOR THE VON KÁRMÁN EQUATIONS A Dissertation Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The Department of Mathematics by Armin Karl Reiser M.S. in Math., Louisiana State University, 2007 Diplom, Eberhard Karls Universität Tübingen, 2008 Erstes Staatsexamen, Eberhard Karls Universität Tübingen, 2008 December 2011

3 To my wife Shelley and our precious son Oliver. ii

4 Acknowledgments First of all, I would like to thank my advisor Dr. Susanne C. Brenner, and Dr. Li-yeng Sung for their mathematical guidance and for their contributions to my work. I would also like to thank Dr. Michael Neilan for insightful discussions on the Monge-Ampère equation and the von Kármán problem, and for his introduction to the COMSOL Software package. All of this was a big stepping stone in my research. Sincere thanks also to my friend Sean Farley, who helped me with his computing expertise in many stages of my work. I express my deepest gratitude to the Institute of Mathematics and its Applications in Minneapolis for my experience as a student in the fall of A substantial part of my research was done in this productive environment. Furthermore, I want to thank the National Science Foundation for providing me summer support under Grant No. DMS Last but not least, I would like to thank my family for their endless support. To my parents, for their help in all thinkable situations of life, making my studies abroad in the United States possible. And to my loving wife Shelley; without her support, none of this would have been possible. She was there when I needed her most. iii

5 Table of Contents Acknowledgments Abstract iii vi 1 Introduction Von Kármán Equations Overview on Finite Element Methods for the von Kármán Equations 4 2 Preliminaries Spectral Theory of Compact Operators on Hilbert Spaces Representation Theorems on Hilbert Spaces Weak Convergence Sobolev Spaces Theoretical Results on the von Kármán Equations The Biharmonic Problem The von Kármán Equations Existence of Solutions Uniqueness and Non-Uniqueness of Solutions The Linearized von Kármán Equations Regularity Results For the Biharmonic Equation For the von Kármán Equations For the Linearized von Kármán Equations Finite Element Methods for the Biharmonic Problem Conforming Finite Element Methods A C 0 Interior Penalty Method for the Biharmonic Problem Notation and Underlying Finite Element Space Derivation of the Method and Error Analysis A C 0 Interior Penalty Method for the von Kármán Equations A C 0 Interior Penalty Method for the Linearized von Kármán Equations C 0 Interior Penalty Methods for the von Kármán Equations Numerical Examples On the Unit Disk On a Square and a L-Shaped Domain iv

6 References Appendix: COMSOL Code Vita v

7 Abstract In this dissertation we develop a C 0 interior penalty method for the von Kármán equations for nonlinear elastic plates. We begin with a brief survey on frequently used finite element methods for the von Kármán equations. After addressing some topics from functional analysis in the preliminaries, we present existence, uniqueness and regularity results for the solutions of the von Kármán equations in Chapter 3. In the next chapter we review the C 0 interior penalty method for the biharmonic problem. Motivated by these results, we propose a C 0 interior penalty method for the linearized von Kármán equations in Chapter 5 and show the wellposedness and stability of this method. We then introduce the new C 0 interior penalty method for von Kármán equations, and establish the corresponding a priori error estimate by a fixed point argument. Numerical examples are presented that confirm the theoretical results. vi

8 Chapter 1 Introduction 1.1 Von Kármán Equations The von Kármán equations are two coupled partial differential equations of fourth order that describe the behavior of an elastic plate of thickness δ subject to transversal loads and lateral forces. In this model, the geometry of the plate P is prescribed by a middle surface. The middle surface of P is a bounded subset R 2 that lies in the (x 1, x 2 ) plane and cuts the plate into two equal halves of thickness δ ; i.e., 2 P = [ δ 2, δ 2 ]. The plate is assumed to be of St. Venant-Kirchhoff material with Lamé constants λ and µ, Young s modulus E, and Poisson ratio ν. The transversal load that acts on the plate is given by a function f δ : R. The resulting transverse displacement is denoted by the function ξ1 δ : R. The stresses on the plate are given in terms of the Airy stress function ξ2 δ : R that relates to the Cauchy stress tensor by σ 11 = 11 ξ δ 2, σ 22 = 22 ξ δ 2, σ 21 = 21 ξ δ 2. If the deflection ξ δ 1 is small relative to the thickness δ, the linear plate theory due to Kirchhoff and Love [36] provides a reasonable approximation for the behavior of the plate. In this model, ξ δ 1 is the solution of the biharmonic equation D 2 ξ δ 1 = f δ, (1.1) where D = Eδ 3 12(1 ν 2 ) 1

9 is the flexural rigidity and 2 is the biharmonic operator, given by 2 u = 1111 u u u. In 1910, Theodore von Kármán [39] observed that the existing theories for plates were limited to the case of small deflections. To capture the behavior of the plate for large deflections more accurately, von Kármán [39] included in his model terms of higher order for the strain tensor. In this way, he obtained the following system of partial differential equations D 2 ξ δ 1 = δ[ξ δ 1, ξ δ 2] + f δ in, (1.2a) 2 ξ δ 2 = E 2 [ξδ 1, ξ δ 1] in, (1.2b) where [, ] is the Monge-Ampère form defined by [u, v] = 11 u 22 v + 22 u 11 v 2 12 u 12 v. Originally, von Kármán did not specify boundary conditions for the equations, and the physical reasoning behind these equations remained controversial until 1980, when Ciarlet [13] showed that the two dimensional von Kármán system is the leading term of a formal asymptotic expansion (in terms of the thickness of the plate as the small parameter) of the exact three-dimensional equations of nonlinear elasticity associated with a specific class of boundary conditions. For a clamped plate, the boundary conditions are ξ δ 1 = n ξ δ 1 = 0 on, ξ δ 2 = ψ δ 0, n ξ δ 2 = ψ δ 1 on, (1.2c) (1.2d) where n denotes the normal derivative, and ψ δ 0 and ψ δ 1 are real-valued functions on that are related to the surface forces acting on the lateral face [ δ 2, δ 2 ] of the plate. 2

10 By introducing the functions ξ 1, ξ 2, f, ψ 0, and ψ 1 with 2D 2 ξ1 δ = δe ξ 1, f δ = δe D3/2 f, ξ δ 2 = D δ ξ 2, ψ δ 0 = D δ ψ 0, ψ δ 1 = D δ ψ 1, the von Kármán equations can be written in a form free of any physical constants. These equations are referred to as the canonical von Kármán equations and are stated as follows. Find (ξ 1, ξ 2 ) such that 2 ξ 1 = [ξ 1, ξ 2 ] + f in, (1.3a) 2 ξ 2 = [ξ 1, ξ 1 ] in, (1.3b) ξ 1 = n ξ 1 = 0 on, ξ 2 = ψ 0, n ξ 2 = ψ 1 on. (1.3c) (1.3d) Due to the combination of nonlinear, second-order terms involving the Monge- Ampère form and linear terms of fourth order from the biharmonic operator, the von Kármán system is called a quasi-linear, fourth order problem. Von Kármán did not have solutions for his problem at hand [39], and although widely investigated, exact solutions were only obtained in rare cases for special geometries and boundary conditions [36, 26]. In 1934 Way [41] was celebrated for finding an exact solution for a circular, uniformly loaded plate (see also [26]). Eight years later, Levy [23] used double Fourier series to solve the von Kármán equations exactly on a simply supported, rectangular plate. Although finding an exact solution remained difficult in general, it was shown that solutions of the von Kármán equations exist for arbitrarily shaped plates [21, 24, 16, 14]. Another feature of the von Kármán equations due to their nonlinear nature is that the solutions are not unique in general. If only small forces are applied, Knightly [21] proved that the solutions are unique - the plate behaves in a pre- 3

11 dictable way. For larger loads, the plate starts to buckle: one deformation is realized among several possible distinct solutions of the von Kármán problem. The critical loading that causes the occurrence of bifurcating solution branches is of particular interest [4, 16]. 1.2 Overview on Finite Element Methods for the von Kármán Equations Various finite element methods have been developed for the von Kármán equations. The finite element method with the easiest formulation is the conforming method that is based on the weak form of the von Kármán problem. Find (ξ 1, ξ 2 ) V such that for all (v 1, v 2 ) V, there holds ( ξ 1 v 1 + ξ 2 v 2 [ξ 1, ξ 2 ]v 1 + [ξ 1, ξ 1 ]v 2 )dx = fv 1 dx, (1.4) where V denotes here the Sobolev space (H 2 0()) 2. The von Kármán equation is then discretized by restricting the variational formulation (1.4) to a finite dimensional subspace V h of V. A function (ξ h,1, ξ h,2 ) V h is called a discrete solution of the conforming method if for all (v 1, v 2 ) V h, there holds ( ξ h,1 v 1 + ξ h,2 v 2 [ξ h,1, ξ h,2 ]v 1 + [ξ h,1, ξ h,1 ]v 2 )dx = fv 1 dx. (1.5) In the case that the exact solution of the von Kármán equations is isolated, Brezzi [10] proved the existence and uniqueness of the discrete solution, and derived optimal error bounds for this method. More recently, the convergence of the conforming method was also established for a von Kármán plate with more general boundary conditions where the plate is partly clamped and partly free [15]. Conforming finite element methods for the space (H 2 0()) 2 require the use of C 1 elements, which are rather complicated and have a large number of degrees of freedom [43]. The simplest C 1 element is the Argyris element with 21 degrees of 4

12 freedom and polynomial shape functions of degree 5. This means that 42 variables have to be known to describe the discrete solution locally on one finite element. Consequently, the size of the global discrete problem is very large for the conforming method, even if the triangulation consists of a relatively small number of elements. One class of finite methods that can be used to avoid C 1 elements are mixed finite element methods. If the solution (ξ 1, ξ 2 ) of the von Kármán equations has the regularity (ξ 1, ξ 2 ) (H 3 ()) 2, the von Kármán equations can be written in an alternative weak form on the space V = H 1 0() (H 1 ()) 3, treating the Hessians 2 ξ 1, 2 ξ 2 (H 1 ()) 4 as additional unknowns. This weak formulation is given as follows. Find (ξ 1, ξ 2 ) V V such that a(ξ 1, v) + [ξ 1, ξ 2 ]v dx = fv dx v V, a(ξ 2, v) [ξ 1, ξ 1 ]v dx = 0 v V, where a(, ) and [, ] are defined here for ξ = (ξ, u 11, u 12, u 22 ) V, and v = (v, v 11, v 12, v 22 ) V with u 21 = u 12 and v 21 = v 12 by [ξ, v] = u 11 v 22 + u 22 v 11 2u 12 v 12, a(ξ, v) = ( ) j ξ i v ij + u ij v ij dx + i j i,j i u ij j v dx. Based on this formulation, Miyoshi [28] defines a mixed method for the von Kármán equations. Since functions in V only need to be weakly differentiable once, C 0 elements can be used to discretize V. The discrete problem is then to find (ξ 1, ξ 2 ) V h V h such that a(ξ 1, v) + [ξ 1, ξ 2 ]v dx = fv dx v V h, a(ξ 2, v) [ξ 1, ξ 1 ]v dx = 0 v V h, 5

13 where V h V is the space of piecewise linear polynomials (v, v 11, v 12, v 22 ) with v 0 on. Since a linear polynomial on a triangle is determined by the three values on the vertices, it follows that the discrete solution of the mixed method depends on 24 unknowns per element. Compared to the conforming method with 42 unknowns per element, the size of the discrete problem is significantly smaller for the mixed method. Miyoshi [28] showed existence and convergence of the approximate solution, provided the exact solution of the von Kármán equation is isolated. Mixed methods for the von Kármán equations were further analyzed in [20, 11, 33] for the approximation of bifurcating solution branches. Another approach to the approximation of the von Kármán equations is the hybrid finite element method by Quarteroni [31]. For this method, the variational formulation (1.4) is restricted to a subspace V h H0() H 2 0() 2 defined as follows. A function (v 1, v 2 ) belongs to V h if, for any triangle T in a regular triangulation of, and i = 1, 2 2 v i =0 on T, v i T n v i T is a cubic polynomial on T, is a linear polynomial on T. Since (v 1, v 2 ) V h are not polynomials inside T, the integrals in the formulation (1.5) cannot be evaluated easily in general. This issue can be overcome by discretizing the space V h further with nonconforming Zienkiewicz elements and a space of piecewise polynomials for the approximation of 2 v 1 and 2 v 2. Quarteroni showed that this hybrid method is convergent if the exact solution of the von Kármán equation is isolated [31]. 6

14 All of the methods presented above lead to a discrete nonlinear problem that is typically solved by Newton s method. The complexity of the computations in Newton s method depends heavily on the size of the Jacobian of the discrete problem. For this reason, discretizations with fewer degrees of freedom are preferable for this problem. In this dissertation, we introduce a finite element method for the von Kármán equation with only 12 degrees of freedom per element. This is achieved by using a quadratic Lagrange finite element space V h for the discretization. Our main result, proved in Section 5.2, is that the discrete solution of this method is well defined and converges to the exact solution of the von Kármán equations, provided the exact solution is unique. The price we pay for using this simple finite element space is that the method is nonconforming. That is, V h is not a subspace of the space V = (H0()) 2 2 where the exact solution lives. Hence, the finite element method cannot be obtained by just restricting the weak formulation (1.4) to the finite element space. Instead, one has to modify (1.4) such that is also defined for functions in the nonconforming finite element space V h. This modification can be obtained by considering the integration by parts formulas, that lead to the weak formulation (1.4), on a local scale, element by element. To mitigate the deficiency of V h not being contained in (C 1 ()) 2, we penalize the jumps of the normal derivatives along inter-element boundaries. We call the resulting method the C 0 interior penalty method for the von Kármán equations (see Section 5.2 and Section 4.2 for details). Before discretizing the von Kármán equations in Section 5.2, we study the theory of the von Kármán problem first. Following the outline of the analysis for the von Kármán equations in [14, 16], we establish in Section 3.2 conditions under 7

15 which the von Kármán problem for a uniform loaded plate has a unique solution ξ (H0()) 2 2. Under these conditions, we show that the linearization of the von Kármán problem at ξ and the corresponding dual problem are invertible. After reviewing the C 0 interior penalty method for the biharmonic equation [9] in Section 4.2, we formulate in Section 5.1 a C 0 interior penalty method for the linearized von Kármán equations. With an adaption of an argument from Schatz [34] and using the invertibility of the linearized von Kármán problem, we then prove that this method has a unique solution and derive an a priori error estimate for the approximation. Furthermore, we show with a new argument that this method is stable, which is a key component in the proof of our main result on the convergence of the C 0 interior penalty method for the von Kármán equations in Section 5.2. We conclude this dissertation with numerical experiments of the new interior penalty method for the von Kármán equations. The theoretically expected rates of convergence are confirmed by our computational results. 8

16 Chapter 2 Preliminaries 2.1 Spectral Theory of Compact Operators on Hilbert Spaces In this section, we review some facts from spectral theory of compact operators on Hilbert spaces. We begin with the definition of a compact operator and the definition of the spectrum. Definition 2.1. Let X, Y be Banach spaces. A bounded linear operator T : X Y is compact if, for every bounded sequence {x n } in X, the sequence {T x n } in Y has a convergent subsequence. Definition 2.2. Let T : X X be a bounded linear operator. The spectrum σ(t ) of T is defined by σ(t ) = {λ C λi T is not bijective}. A complex number λ C is said to be an eigenvalue of T, if λi T is not injective, or equivalently if T x = λx for some 0 x H. Remark 2.3. If λ σ(t ), then the Inverse Mapping Theorem [32] implies that the resolvent (λi T ) 1 is a bounded linear operator. Clearly, every eigenvalue of a bounded linear operator T is contained in the spectrum of T. More can be said about the spectrum of a compact operator. Theorem 2.4. (Riesz-Schauder Theorem, [32]) Let H be a Hilbert space. The spectrum of a compact operator T : H H is discrete with no limit points other than possibly 0. Moreover, any nonzero λ σ(t ) is an eigenvalue of T with finite multiplicity. 9

17 Definition 2.5. Let H be a Hilbert space with inner product (, ). A linear operator T : H H is called symmetric if (T x, y) = (x, T y) x, y H. A standard result in spectral theory is that the eigenvalues of a symmetric operator are real-valued [25]. A more precise statement on the distribution of the eigenvalues holds for a compact, symmetric operator (see [25], for instance). Theorem 2.6. Let H be a Hilbert space with scalar product (, ) and T be a symmetric, compact, linear operator on H. Then M = sup x H x 0 (T x, x), and m = inf (x, x) x H x 0 belong to the spectrum σ(t ), and σ(t ) [m, M]. (T x, x) (x, x) (2.1) Definition 2.7. Let H be a Hilbert space with inner product (, ). A linear operator T : H H is called positive definite if (T x, x) > 0 0 x H. From Theorem 2.4 and Theorem 2.6, we conclude the following for the eigenvalues of a positive definite operator. Corollary 2.8. Let T : H H be a compact, symmetric, positive definite operator on a Hilbert space H. Then M defined in (2.1) is the largest eigenvalue of T. Moreover, the eigenvalues of T form a non-increasing sequence {λ k } (0, M] that is either finite, or converges to Representation Theorems on Hilbert Spaces This section covers the solvability of the abstract problem below. Given f H, find u H such that a(u, v) = f(v) v H, 10

18 where a(, ) : H H R is a bounded bilinear form. That is, there exists C > 0 such that a(u, v) C u H v H u, v H. (2.2) For the special case that a(, ) is the inner product associated with H, the Riesz- Representation Theorem [32] states the following. Theorem 2.9. (Riesz-Representation Theorem) Let H be a Hilbert space with inner product (, ). Then, for any f H, there exists a unique u H with u H = f H such that (u, v) = f(v) v H. Suppose now that the bounded bilinear form a(, ) is also coercive; i.e., there exists a constant α > 0 such that a(v, v) α v 2 H v H. (2.3) If additionally a(, ) is symmetric, it follows that a(, ) is itself an inner product on H. By (2.2) and (2.3), we have that α v 2 H a(v, v) C v 2 H v H. (2.4) Consequently, the inner product a(, ) induces a norm equivalent to the standard norm on H, and we can apply Theorem 2.9. Corollary Let H be a Hilbert space and a(, ) be a symmetric bilinear form on H H. If a(, ) is bounded and coercive, then a(, ) forms an inner product on H. Its induced norm is equivalent to the standard norm in H, and for every f H, there exists a unique u H such that a(u, v) = f(v) v H. (2.5) 11

19 The next theorem shows that the symmetry of a(, ) is not necessary for the representation (2.5) to hold. A proof can be found in [18, 8, 12]. Theorem (Lax-Milgram Lemma) Let H be a Hilbert space and a(, ) be a bounded bilinear form on H H. If a(, ) is coercive with coercivity constant α > 0, then for any f H, there exists a unique u H such that a(u, v) = f(v) v H. (2.6) Remark If u H is the unique solution of (2.6), then α u 2 H a(u, u) = f(u) f H u H. This implies the stability estimate u H 1 α f H. 2.3 Weak Convergence In this section, we introduce the notion of weak convergence in a Hilbert space H. It is a meaningful tool in many compactness arguments. Definition Let {x n } be a sequence in a Hilbert space H, and x H. We say x n x converges weakly if (x n, y) (x, y) y H. Remark By the Cauchy-Schwarz inequality, it follows that strong convergence implies weak convergence. Lemma Let H be a Hilbert space. If x k x converges strongly in H and y k y converges weakly in H, then (x k, y k ) (x, y). Proof. Since (y k, z) is convergent for any z H, the sequence {(y k, z)} is bounded for every z H. By the principle of uniform boundedness [32], there exists M > 0 12

20 such that y k H < M for all k N. Hence, as k, we have that (x k, y k ) (x, y) = (x, y k y) + (x k x, y k ) (x, y k y) + M x k x H 0. Lemma (Sub-Subsequence Lemma) Let {x k } be a sequence in a Hilbert space H, and x H. Then the following is equivalent. (i) The sequence {x k } converges strongly to x. (ii) Every subsequence of {x k } has a subsequence that converges to x. Proof. The implication (i) (ii) is obvious. For the other direction, suppose {x k } does not converge to x. Then we can find ɛ > 0 such that for any k N there exists k 0 > k with x k0 x H ɛ. In the same manner, we find k 1 > k 0 with x k1 x H ɛ. Repeating this argument, we obtain a subsequence {x kn } that satisfies x kn x H ɛ for all n N. Clearly, the sequence {x kn } does not have a subsequence converging to x. Lemma Let H be a Hilbert space that is compactly embedded into a Hilbert space X. If x k x weakly in H, then x k x strongly in X. Proof. As seen in the proof of Lemma 2.15, it follows from the principle of uniform boundedness that the sequence {x k } is bounded in H. Now let {x l } be an arbitrary subsequence of {x k }. Since H is compactly embedded in X, the sequence {x l } has a subsequence {x m } that converges strongly in X, say x m y. It follows that x m y in X. But it also holds that x m x in H, because {x m } is a subsequence of {x k }. Since the weak limit is unique in H, we conclude that y = x. Hence, x m x strongly in X. The Sub-Subsequence Lemma completes the proof. 13

21 The next lemma can be seen as a compactness result in the topology of weak convergence. Its proof involves a diagonal argument (cf. [42, Theorem III.3.7]). Lemma Every bounded sequence in a Hilbert space H has a weakly convergent subsequence. Definition Let H be a Hilbert space. A functional f : H R is called weakly lower semicontinuous, if f(x) lim inf f(x k ) whenever x k x. (2.7) Example Let H be a Hilbert space. Then the functional f(x) = x H is weakly lower semicontinuous. This follows from the observation that x 2 H = (x, x) = lim k (x, x k ) = lim inf(x, x k ) lim inf x H x k H as x k x. k k Definition Let H be a Hilbert space. We call f H coercive, if f(x) as x H. The final theorem of this section is a classical result from the direct methods in the calculus of variations [35]. Theorem Let H be a Hilbert space, and f H be a weakly lower semicontinuous, coercive functional. Then f is bounded below, and there exists x H such that f(x) = inf f(y). (2.8) y H Proof. If f is not bounded from below, there is a sequence {x k } in H such that f(x k ). From the coercivity of f, it follows that {x k } is bounded. By Lemma 2.18, {x k } has a convergent subsequence, say x kl x for some x H. Then the weakly lower semicontinuity of f implies f(x) lim inf l f(x kl ) =. 14

22 Therefore, f is bounded below. Let {x k } be a minimizing sequence in H, i.e., f(x k ) inf f(y). (2.9) y H As {x k } is bounded by coercivity, we can find by Lemma 2.18 a subsequence {x kl } that converges weakly to some x H. From (2.9) and (2.7), it follows that This concludes the theorem. f(x) lim inf l 2.4 Sobolev Spaces f(x kl ) = inf y H f(y). Let be a bounded open subset in R n. The space of infinitely differentiable functions on with compact support is denoted by C c (), and the space of locally integrable functions is defined by L 1 loc() = {f : R f L 1 (D) for any compact subset D }. A function f L 1 loc () is weakly differentiable of order α if there exists a function g L 1 loc () such that f α v dx = ( 1) α gv dx v C c (), (2.10) where α is a vector (α 1, α 2,..., α n ) of nonnegative integers with α = n i=1 α i, and α f denotes the partial derivatives in the multiindex notation ( x 1 ) α1 ( x n ) αn f. We call the function g in (2.10) the α th weak derivative of f, and write α f for g. For the weak derivatives (2,0) f, (0,2) f, and (1,1) f, we often use the notation 11 f, 22 f, and 12 f. Definition (Sobolev Spaces, [1, 18, 37]) For an integer k 0 and 1 p, the Sobolev space W k p () is defined by W k p () = {f L 1 loc() α f L p () α k}. 15

23 The Sobolev space Wp k () is equipped with the norm ( 1/p α k f W k p () = α f p L ()) for 1 p < p max α k α f L () for p =. and the seminorm ( 1/p α =k f W k p () = α f p L ()) for 1 p < p. max α =k α f L () for p =. For s > 0 with s = k + σ for some nonnegative integer k and some 0 < σ < 1, the fractional Sobolev space Wp s () is defined for 1 p < by W s p () = {f W k p () f W s p () < }, where f p W s p () = f p W k p () + α =m ( ) α f(x) α f(y) p dxdy. (2.11) x y n+σp Remark One can also define the fractional Sobolev space W s p () by means of interpolation between W k p () and W k+1 p () (see [1, 37, 38]). Theorem The space W s p () is a Banach space. For p = 2, the Sobolev space W s p () is a Hilbert space, and we refer to it by H s (). For k N, the inner product on H k () is given by (f, g) H k () = α k α f α g dx. The next theorem shows that functions in a Sobolev space can be approximated by smooth functions. A proof of this theorem can be found in [18, 37, 1]. Theorem Let 1 p <, and k be a nonnegative integer. Then C () is dense in W k p (). 16

24 Definition Let s > 0. The closure of C c () in H s () is denoted by H s 0(). Definition The dual space of H s 0() is called H s (), and its norm is defined for F H s () by F H s () = sup v H s 0 () v 0 F (v). (2.12) v H s () Remark Since the dual space of H s () is isometric to H s 0(), the norm of u H 2 0() can be calculated with the formula u H s () = sup F H s () F 0 Theorem (Sobolev Embedding Theorem, [38]) F (u). (2.13) F H s () Let be an arbitrary domain in R n, let m 0 and t 0, and 1 < p <. Then the following continuous embeddings exist. (i) W m+t p () C t (), provided m > n p, (ii) W m+t p () W t q(), provided m n p n q Theorem (Rellich-Kondrachov, [1]) and p q <. Let be a bounded domain in R n with piecewise smooth boundary, let t 0 and k 1 be integers, and 1 p <. Then the following embeddings are compact. (i) W k+t p () W t q() with 1 q < np n kp, provided k < n p. (ii) W k+t p () W t q() with 1 q < provided k n p. (iii) W k+t p () C t (), provided k > n p. Corollary Let be a bounded domain with piecewise smooth boundary. Then L 1 () is continuously embedded into H 2 (). Proof. By the Sobolev embedding theorem (Theorem 2.30), there exists a continuous embedding H 2 () C 0 (), that is, there exists C > 0 such that φ C φ H 2 () φ C c (). 17

25 For g L 1 (), we have g H 2 () = sup φ Cc () φ 0 g, φ φ H 2 () g L 1 () φ φ H 2 () C g L 1 (), where, denotes here the duality between H 2 0() and H 2 (). Theorem (Trace Theorem, [38]) Let be a polygon with piecewise smooth boundary, s > 1, and k be the largest 2 integer smaller than s 1. Then there exists a bounded linear bounded operator 2 TR from H s () onto k l=0 Hs 1 2 l ( ) such that ( TR v = v, n v k ),, n v k where n is the outer normal with respect to. v C c (), With the bounded linear operator TR, the space H s 0() can be characterized as follows (see [38]). Lemma Under the assumptions of Theorem 2.33, the following holds H s 0() = {v H s () TR v = 0}. Theorem (Poincaré s Inequality, [1]) Let R n be a bounded domain. Then there exists a constant K > 0 such that v L 2 () K v H 1 () v C c (). (2.14) Since H0() 1 is the closure of Cc () with respect to H 1 (), inequality (2.14) also holds for all v H0(). 1 Then it is easy to see that on H0() 1 the seminorm H 1 () turns out to be a norm equivalent to H 1 (). By applying Poincaré s inequality successively on derivatives of higher order, we obtain the following. Corollary Let R n be a bounded domain, and k be a positive integer. Then H k () is a norm on H0 k () equivalent to the norm H (). k 18

26 Remark To avoid the accumulation of constants, we will sometimes use the notation A B indicating an inequality of the type A C B, where C is a positive generic constant, independent from any discretization parameters. If this relation holds in both directions, we write A B. Thus, Corollary 2.36 becomes H k () H k (). (2.15) 19

27 Chapter 3 Theoretical Results on the von Kármán Equations As the biharmonic operator is an integral part of the von Kármán problem, we begin this chapter with a section on the biharmonic problem. 3.1 The Biharmonic Problem On a bounded domain R 2, we consider the biharmonic problem (1.1) with homogeneous boundary conditions, that is, 2 u =f in, (3.1a) u = n u = 0 on. (3.1b) If u H 4 () H0(), 2 integration by parts (see [12]) yields fv dx = 2 u v dx = u v dx v H0(). 2 (3.2) This motivates the weak formulation of the biharmonic problem. For F H 2 (), find u H0() 2 such that (u, v) = F (v) v H 2 0(), (3.3) where (u, v) is the symmetric bilinear form (u, v) = u v dx. (3.4) For v Cc (), integration by parts [12] yields (v, v) = v 2 L 2 () = = ( ( ii v) 2 + ii v jj v ) dx i i j ( ( ii v) 2 + ( ij v) 2) dx = v 2 H 2 (). (3.5) i i j 20

28 Thus, Corollary 2.36 implies that (, ) is an inner product on H 2 0(). Moreover, v = (v, v) defines a norm on H 2 0() that coincides with H 2 (). Hence, the norm is equivalent to the Sobolev norm H 2 () on H 2 0(); i.e., v v H 2 () v H 2 0(). (3.6) From the above, we conclude that (, ) is a coercive, bounded bilinear form on H0() 2 H0(). 2 Furthermore, Theorem 2.9 implies the following. Theorem 3.1. For any F H 2 (), the biharmonic problem (3.3) has a unique solution u H0(). 2 Moreover, there exists C > 0 such that u H 2 () C f H 2 (). (3.7) An alternative way of carrying out the integration by parts in (3.2) is 2 fv dx = 2 u v dx = ( u) v dx = i=1 2 2 = i u i v dx = ij u ij v dx. i=1 i,j=1 This leads to another weak formulation of the biharmonic problem. Given F H 2 (), find u H 2 0() such that ( i u) i v dx a(u, v) = F (v), (3.8) where a(, ) : V V R is the symmetric bilinear form defined by 2 a(w, v) = ij w ij v dx. (3.9) i,j=1 Remark 3.2. By Lemma 2.11, there exists a unique solution of (3.8). Moreover, one can show that the unique solutions of the weak problems (3.3) and (3.8) coincide. 21

29 3.2 The von Kármán Equations Having introduced the biharmonic problem and its underlying Sobolev spaces, we proceed with theoretical considerations on the von Kármán equations. The analysis here essentially follows Ciarlet [14]. First, let us endow the von Kármán equations (1.3) with the appropriate function spaces from Section 2.4. Interpreting the equations (1.3) in the weak sense, the von Kármán problem is to find (ξ, ψ) H0() 2 H 2 () such that 2 ξ = [ξ, ψ] + f in, (3.10a) 2 ψ = [ξ, ξ] in, (3.10b) ψ = ψ 0, n ψ = ψ 1 on. (3.10c) where f H 2 (), ψ 0 H 3/2 ( ), and ψ 1 H 1/2 (). Note that the boundary condition (1.3c) is enforced by the space H0(). 2 Moreover, the spaces H 3/2 () and H 1/2 () are by the trace theorem (Theorem 2.33) the appropriate function spaces for the boundary conditions (3.10c). To formulate the von Kármán problem on H0() H 2 0(), 2 we decompose ψ into ψ = ξ 2 + θ, where ξ 2 H 2 0(), and θ H 2 () is the unique solution of 2 θ = 0 in, θ = ψ 0, n θ = ψ 1 on. (3.11a) (3.11b) Writing ξ 1 for ξ and replacing ψ by θ + ξ 2, we obtain the following version of the von Kármán equations. 22

30 Given f H 2 (), find ξ = (ξ 1, ξ 2 ) H 2 0() H 2 0() such that 2 ξ 1 = [ξ 1, ξ 2 ] + [ξ 1, θ] + f in, (3.12a) 2 ξ 2 = [ξ 1, ξ 1 ] in. (3.12b) where θ is defined by (3.11). For the remainder of this thesis, we use this formulation of the von Kármán equations, keeping in mind that the Airy stress function of the plate is actually ψ = θ + ξ 2. Remark 3.3. To simplify the notation in the product spaces that arise in the von Kármán problem, we frequently use the boldface style for elements in a product space: We denote elements in the product space X X = X 2 of a Banach space X with norm X by x = (x 1, x 2 ). The product norm of x is then given by x 2 X = x X + x 2 2 X. By Theorem 3.1 the two biharmonic equations (3.12a) and (3.12b) of the von Kármán system are well defined, if the corresponding right hand sides, in particular the terms with the Monge-Ampère form, are in H 2 (). As we will see now, this is indeed the case. For η, χ H 2 (), the Cauchy-Schwarz inequality yields ( [η, χ] L 1 () 11 η 22 χ + 22 η 11 χ η 12 χ ) dx χ H 2 () η H 2 (). (3.13) The continuous embedding L 1 () H 2 () of Corollary 2.32 then implies that [η, χ] H 2 (). In other words, the Monge-Ampère form maps functions from H 2 () H 2 () to H 2 (). Thus, the following operator is defined by Theorem 3.1. Definition 3.4. Define the operator B : H 2 () H 2 () H0() 2 that maps (η, χ) to the unique solution B(η, χ) H0() 2 of 2 B(η, χ) = [η, χ]. 23

31 Remark 3.5. The operator B(, ) is symmetric and bilinear. This follows directly from the linearity of the biharmonic operator and the fact that the Monge-Ampère form is symmetric and bilinear. The operator B(, ) reduces the von Kármán equations to a single equation in H0(), 2 the reduced von Kármán equation (see [14]). Theorem 3.6. For f H 2 (), let f H 2 0 () be the unique solution of the biharmonic problem 2 f = f, let θ be defined by (3.11), and C be the operator on H0() 2 defined by Cu = B(B(u, u), u). (3.14) Then (ξ 1, ξ 2 ) (H0()) 2 2 is a solution of the von Kármán equations (3.12) if and only if ξ 1 H0() 2 satisifies the reduced von Kármán equation ξ 1 + Cξ 1 B(θ, ξ 1 ) + f = 0, (3.15) and ξ 2 is determined by ξ 1 via ξ 2 = B(ξ 1, ξ 1 ). (3.16) Proof. The equivalence of (3.12b) and (3.16) follows immediately from the definition of B(, ). Thus, ξ 1 is a solution of the von Kármán equations (3.12) if and only if ξ 1 satisfies 2 ξ 1 = [ξ 1, B(ξ 1, ξ 1 )] + [θ, ξ 1 ] + f. By Definition 3.4 and the definition of f, this is equivalent to ξ 1 = B(ξ 1, B(ξ 1, ξ 1 )) + B(θ, ξ 1 ) + f. This proves the theorem. In the next lemma, we summarize the main properties of the Monge-Ampère form for the analysis of the von Kármán equations [14]. 24

32 Lemma 3.7. The mappings (u, v, w) [u, v]w dx, (3.17) ( b: (u, v, w) 12 u( 1 v 2 w + 2 v 1 w) ( 11 u 2 v 2 w + 22 u 1 v 1 w) ) dx (3.18) are bounded trilinear forms on H 2 () H 2 () H 2 () with the following properties. (i) If one of the arguments u, v, w belongs to H0(), 2 then [u, v]w dx = b(u, v, w) = [u, w]v dx. (3.19) (ii) There exists C > 0 such that [u, v]w dx C u H 2 () v W 1 4 () w W 1 4 (). (3.20) (iii) If u W 2 (), then there exists C > 0 such that b(u, v, w) u W 2 () v H 1 () w H 1 (). (3.21) Proof. Due to the Sobolev embedding H 2 () C 0 () (see Theorem 2.30), there exists C > 0 such that w L () C w H 2 (). Thus, we can estimate using (3.13) [u, v]w dx w L () [u, v] L 1 () C u H 2 () v H 2 () w H 2 (). We conclude that the mapping (3.17) is a continuous trilinear form. In a similar way, one can show that the trilinear form b(,, ) is continuous. The Sobolev embedding H 2 () W 1 4 () (see Theorem 2.30) and the Cauchy-Schwarz inequality yield b(u, v, w) u H 2 () 1 i,j 2 ( ) 1 i v j w 2 2 dx u H 2 () v W 1 4 () w W 1 4 () (3.22) u H 2 () v H 2 () w H 2 (). 25

33 Since both trilinear forms are continuous, it suffices to show that equality (3.19) holds for functions in a dense subspace of H 2 (), with one of the arguments belonging to a dense subspace in H 2 0(). Let u, v, w C () and suppose one of the three functions has compact support. By integration by parts, we verify that [u, v]w dx = ( 11 v 22 u)w + ( 22 v 11 u)w 2( 12 u 12 v)w dx = 1 v 1 (w 22 u) dx 2 v 2 (w 11 u) dx + 2 v 1 (w 12 u) dx + 1 v 2 (w 12 u) dx = 12 u( 1 v 2 w + 2 v 1 w) ( 11 u 2 v 2 w + 22 u 1 v 1 w) dx = b(u, v, w). In consideration of Theorem 2.26 and Definition 2.27, this implies the first equality in (3.19). The second equality in (3.19) is then a consequence of the fact that b(u, v, w) is symmetric with respect v and w. Moreover, inequality (3.20) follows from (3.19) and estimate (3.22). Finally, if u W () 2 we can estimate 2 b(u, v, w) max { iju L i,j ()} i v j w dx u W 2 v H 1 () w H 1 (). i,j=1 Corollary 3.8. Let u H 2 () and v, w H0(). 2 Then (B(u, v), w) = (B(u, w), v). Proof. By (3.19) and Definition 3.4, we have (B(u, v), w) = B(u, v) w dx = [u, v]w dx = [u, w]v dx = B(u, w) v dx = (B(u, w), v). 26

34 With the previous lemma and the associated corollary one can derive the main properties of the operators in the reduced von Kármán equation (cf. [14]). Lemma 3.9. Let θ H 2 () and B(, ) be the operator from Definition 3.4. Then the following holds. (i) The operator B : H 2 0() H 2 0() H 2 0() is bounded; i.e., there exists C > 0 such that B(u, v) C u v u, v H 2 0(). (3.23) Moreover, B is sequentially compact; i.e., if (u k, v k ) (u, v) converges weakly in H 2 () H 2 (), then B(u k, v k ) B(u, v) converges strongly in H 2 0(). (ii) The operator C : H 2 0() H 2 0() defined in (3.14) is sequentially compact and positive definite in the sense that (Cu, u) > 0 0 u H 2 0(). (iii) The mapping u B(θ, u) defines a compact operator on H 2 0() that is symmetric with respect to (, ). Proof. By (3.20) and (3.6), we have for u, v H 2 () B(u, v) = sup w H 2 0 () w 0 = sup w H 2 0 () w 0 (B(u, v), w) w [u, v]w dx w u W 1 4 () v W 1 4 (). (3.24) 27

35 Thus, the Sobolev embedding H 2 () W 4 1 () together with (3.6) imply (3.23). Furthermore, we obtain from (3.24) that B(u k,v k ) B(u, v) B(u k u, v) + B(u, v k v) + B(u k u, v k v) ( u k u W 1 4 () + v k v W 1 4 () + u k u W 1 4 () v k v W 1 4 ()). (3.25) Since (u k, v k ) (u, v) in H 2 () H 2 (), and since H 2 () is compactly embedded in W 1 4 () (see Theorem 2.31), Lemma 2.17 implies that (u k, v k ) (u, v) converges strongly in W 1 4 () W 1 4 (). This together with estimate (3.25) proves (i). Let u H 2 0(). Since B(u, u) H 2 0(), we obtain from Corollary 3.8 that (Cu, u) = (B(B(u, u), u), u) = (B(u, u), B(u, u)) 0. (3.26) Moreover, it follows that (Cu, u) = 0 if and only if B(u, u) = 0. By Definition 3.4, this is equivalent to [u, u] = 0. Thus, for the positive definiteness of C, we only need to show that [u, u] = 0 implies u = 0. Consider the function v H 2 () defined by v(x 1, x 2 ) = 1 2 (x2 1 + x 2 2). Then [u, v] = 11 u 22 v + 22 u 11 v = u. If [u, u] = 0, integration by parts yields together with identity (3.19) that 0 = [u, u]v dx = [u, v]u dx = ( u)u dx = u u dx = u 2 H 1 (). Therefore, u is constant on. From the fact that u = 0 on (cf. Lemma 2.34), we conclude that u = 0. Since the sequential compactness of C is a direct consequence of (i), this establishes (ii). The symmetry of the operator u B(θ, u) follows directly from Corollary 3.8. To show compactness, let {u k } be a bounded sequence in H 2 0(). By Lemma 2.18, 28

36 {u k } has a weakly convergent subsequence, say {u kl }. Because of the sequential compactness of B, the sequence {B(θ, u kl )} is a convergent subsequence of {B(θ, u k )}. Thus, by Definition 2.1, the map u B(θ, u) is a compact operator. This completes the proof. In the preceding lemma, we have seen that the operator B(, ) is bounded in the sense of (3.23). Therefore, we can define B = sup u, v H 2 0 () u 0, v 0 B(u, v) u v. (3.27) The boundedness of B(, ) allows us to investigate the operator u Cu with respect to perturbations in u. The next lemma results from an exercise in [14]. Lemma Let u, v H 2 0(). Then Cu Cv 3 B 2 u v max{ u 2, v 2 }. (3.28) Proof. Since B(, ) is bilinear and symmetric, we have for any u, v H 2 0() Cu Cv = B(u, B(u, u)) B(v, B(v, v)) = B(u, B(u, u v)) + B(v, B(v, u v)) + B(u v, B(u, v)). Hence, (3.27) and the geometric mean inequality imply Cu Cv B 2 u v ( u 2 + v 2 + u v ) B 2 u v ( u 2 + v u v 2 ) = 3 2 B 2 u v ( u 2 + v 2 ) 3 B 2 u v max{ u 2, v 2 }. 29

37 3.2.1 Existence of Solutions Having derived the essential properties of the operators in the reduced von Kármán equation, we address now the existence of solutions of the von Kármán equations. In the first lemma, which can be found in [14], we see that the solutions of the von Kármán equations correspond to the stationary points of a certain functional. Lemma Suppose the operator C, and the functions θ H 2 () and f H0() 2 are given as in Theorem 3.6. Define the functional j : H0() 2 R by j(u) = 1 4 (Cu, u) (u, u) 1 2 (B(θ, u), u) ( f, u). (3.29) Then j is differentiable on H 2 0(), and ξ 1 H 2 0() is a solution of (3.15), if and only if j (ξ 1 ) = 0. Proof. Let u H 2 0() be arbitrary. As B(, ) is bilinear, bounded and symmetric (see Lemma 3.9), we have B(u + v, u + v) B(u, u) = 2B(u, v) + B(v, v), and B(v, v) v 0 as v 0. Thus, B(, ) is differentiable and its derivative at u is 2B(u, ). Similarly, since (, ) and (B(θ, ), ) are bilinear, bounded and symmetric, it follows that their derivatives at u are 2(u, ) and 2(B(θ, u), ), respectively. Lastly, it follows from the chain rule that the derivative of (B(, ), B(, )) at u is 2(B(u, u), 2B(u, )). Thus, by (3.26), the derivative of (C, ) at u is 4(Cu, ). Combing the calculations above, we obtain for the derivative of j at u the functional j (u) = u + Cu B(θ, u) f. Theorem 3.6 completes the proof. 30

38 In consideration of Lemma 3.11, the existence of solutions for the von Kármán equations can be proved by finding stationary points of the functional j. This is done in the next theorem (see [14]). Theorem Under the conditions in (3.10), there exists a solution of the von Kármán equations (3.12) that globally minimizes the functional j in (3.29). Proof. By Lemma 3.11, it suffices to show that there exists a minimizer ξ 1 H 2 0() of the functional j, i.e., j(ξ 1 ) = inf j(u). u H0 2() By Theorem 2.22, we only need to check that j is weakly lower semicontinuous and coercive. First we prove that j is weakly lower semicontinuous. Suppose u k u converges weakly in H 2 0(). From Example 2.20 we know that u lim inf k u k. Taking the square of this inequality, we obtain that (, ) is weakly lower semicontinuous. Moreover, since C and B(θ, ) are sequentially compact (see Lemma 3.9), Cu k Cu and B(θ, u k ) B(θ, u) converge strongly in H 2 0(). Thus, Lemma 2.15 implies (Cu k, u k ) (Cu, u), (3.30) (B(θ, u k ), u k ) (B(θ, u), u). (3.31) Note that the convergence ( f, u k ) ( f, u) is a direct consequence of the weak convergence u k u in H 2 0(). Therefore, j is weakly lower semicontinuous. Next, suppose j is not coercive. Then by Definition 2.21, there exists a constant M > 0 and an unbounded sequence {u k } in H 2 0() such that j(u k ) M k N. (3.32) 31

39 Without loss of generality, we can assume that u k 0 for all k N. Then j(u k ) + ( f, u k ) u k 2 M + ( f, u k ). (3.33) u k 2 Since the sequence {v k }, defined by v k = u k u k with v k = 1 k N is bounded, there exists by Theorem 2.18 a weakly convergent subsequence of {v k }, say v kl v. Passing inequality (3.33) to the subsequence {u kl } yields (B(θ, v k l ), v kl ) u k l 2 (Cv kl, v kl ) M + ( f, v kl ) l N. (3.34) u kl 2 u kl As in (3.30), the weak convergence v kl v implies (Cv kl, v kl ) (Cv, v). According to Lemma 3.9(ii) there are the following two cases: If (Cv, v) > 0, then the left hand side in (3.34) is unbounded. This contradicts the fact that the right hand side tends to 0 as l. Otherwise, we have that v = 0. Then it follows from (3.31) that (B(θ, v kl ), v kl ) approaches 0 as l. Consequently, the left hand side in (3.34) exceeds 1 2 as l, whereas the right hand side approaches 0. We conclude in both cases that j is coercive Uniqueness and Non-Uniqueness of Solutions Although a solution of the von Kármán equations always exists (see Theorem 3.12), the solution is not necessarily unique. In this section, we determine conditions for the uniqueness of solutions of the von Kármán plate (3.10) subject to uniform loading along the lateral faces. The analysis here differs only slightly from [14]. The boundary conditions that correspond to the uniform lateral loading are ψ 0 = p 2 (x2 1 + x 2 2) and ψ 1 = p 2 n (x2 1 + x 2 2), (3.35) 32

40 where p R is a parameter proportional to the magnitude of the lateral force. For this special choice of boundary conditions, θ(x 1, x 2 ) = p 2 (x2 1 + x 2 2) H 2 () (3.36) is the unique solution of (3.11). The Monge-Ampère form [θ, u] for u H0() 2 reduces then to [θ, u] = p u. (3.37) Hence, the von Kármán equations (3.12) can be stated as follows. Given f H 2 (), find (ξ 1, ξ 2 ) H0() 2 H0() 2 such that 2 ξ 1 = [ξ 1, ξ 2 ] p ξ 1 + f in, (3.38a) 2 ξ 2 = [ξ 1, ξ 1 ] in. (3.38b) Likewise, the reduced von Kármán equation (3.15) reduces to the following problem. Find ξ 1 H 2 0() such that where Λ: H 2 0() H 2 0() is the operator defined by Cξ 1 + ξ 1 pλξ 1 f = 0, (3.39) Λu := 1 B(θ, u). (3.40) p As we will see in this section, the uniqueness of the von Kármán equations is closely linked to the spectral properties of Λ. Lemma The operator Λ : H 2 0() H 2 0() is a compact and symmetric positive definite. Proof. For u H0(), 2 integration by parts together with (3.37) implies (Λu, u) = 1 p (B(θ, u), u) = 1 B(θ, u) u dx p = 1 [θ, u]u dx = u u dx = ( u) 2 dx = u 2 H p 1 () 0. 33

41 Since H 1 () is a norm on H 1 0() and since H 2 0() H 1 0(), this implies that (Λu, u) = 0 if and only if u = 0. Thus, Λ is positive definite. The remaining properties of Λ follow directly from the properties of the operator B(θ, ) in Lemma 3.9(iii). As Λ is symmetric, positive definite, and compact, we obtain from Corollary 2.8 that the largest eigenvalue λ 1 of Λ is given by λ 1 = sup u H 2 0 () u 0 (Λu, u) (u, u). Thus, we can define p 1 = 1 λ 1 = inf u H 2 0 () u 0 (u, u) (Λu, u). (3.41) This constant determines if the homogeneous von Kármán problem (3.38) has unique or non-unique solutions, as we see in the next theorem from [14]. Theorem Let f = 0 and p 1 > 0 be the constant defined in (3.41). Then the following holds. (i) If p p 1, then ξ 1 = 0 is the unique solution of problem (3.39). (ii) If p > p 1, then problem (3.39) has at least three solutions: Besides ξ 1 = 0, there also exist two nontrivial solutions ξ 1 H 2 0() and ξ 1 H 2 0(). Proof. For the proof of (i), we refer to Remark To show (ii), consider the normalized eigenvector u H 2 0() with u 2 = 1 that corresponds to the largest eigenvalue 1 p 1 of Λ. By (3.29) we have j(αu) = α4 4 (Cu, u) + α2 2 (1 p p 1 ). Since 1 p p 1 is negative for p > p 1, it follows that j(αu) < 0 for some α sufficiently small. Consequently, the solution ξ 1 = 0 is not the global minimizer of the 34

42 functional j. Hence, by Theorem 3.12, there exists another solution ξ 1 that minimizes the functional j globally. In view of the fact that C is cubic and Λ is linear, it is clear that ξ 1 is another solution of the homogenous reduced von Kármán equation (3.39). In the next lemma, we show a stability estimate for the reduced von Kármán equation. The case p 0 of this lemma is stated as an exercise in [14]. Lemma Let p < p 1. Then the following holds. (i) Any u H0() 2 satisfies min{1, 1 p } u 2 u 2 p(λu, u). (3.42) p 1 (ii) If ξ 1 H 2 0() is a solution of the reduced von Kármán equation (3.39), then ξ 1 max{1, Proof. If p 0, then it follows from (3.41) that p 1 p 1 p } f. (3.43) p(λξ 1, ξ 1 ) p p 1 ξ 1 2. Consequently, we have (1 p p 1 ) u 2 u 2 p(λu, u). Otherwise, if p < 0, then the fact that Λ is a positive definite operator (see Lemma 3.13) implies u 2 u 2 p(λu, u). This proves statement (i). Suppose ξ 1 H0() 2 is a solution of the reduced von Kármán equation (3.39). Then ξ (Cξ 1, ξ 1 ) p(λξ 1, ξ 1 ) ( f, ξ 1 ) = 0. (3.44) 35

43 Using the result of statement (i), we obtain from this the estimate min{1, 1 p p 1 } u 2 u 2 p(λu, u) = ( f, ξ 1 ) (Cξ 1, ξ 1 ). (3.45) By Lemma 3.9(ii), (Cξ 1, ξ 1 ) is nonnegative. Thus, (3.45) and the Cauchy-Schwarz inequality yield min{1, 1 p p 1 } u 2 ( f, ξ 1 ) f ξ 1. Dividing both sides by min{1, 1 p p 1 } leads to ξ 1 2 max{1, p 1 p 1 p } f ξ 1. The fact that, for f 0, the solution ξ 1 of the reduced von Kármán equation (3.39) is nontrivial completes the proof. Remark The statement (ii) of this lemma proves Theorem 3.14 for p p 1. In the next theorem, we show that the von Kármán equations (3.38) has a unique solution if certain conditions on p and f hold. Similar versions of this theorem can be found in [14, 16]. Theorem Suppose f H0() 2 is sufficiently small such that 3 B 2 f 2 < 1. (3.46) Moreover, assume that p < p 1 (1 3 3 B 2 3 f 2 3 ). (3.47) Then the solution of the reduced von Kármán equation (3.39) is unique. Proof. First, let us bring the two conditions (3.47) and (3.46) into an easier form that is advantageous for our estimates. The assumption (3.47) can be written as p 1 p > 3 3 B 2/3 2/3 f p. 1 36