Dedicated to Ivo Marek on the occasion of his 75th birthday. T(λ)x = 0 (1.1)

Transcription

1 A MINMAX PRINCIPLE FOR NONLINEAR EIGENPROBLEMS DEPENDING CONTINUOUSLY ON THE EIGENPARAMETER HEINRICH VOSS Key words. nonlinear eigenvalue problem, variational characterization, minmax principle, variational eigenproblem, vibration of plates with masses AMS subject classification. 35P30,65F15 Abstract. Variational characterizations of real eigenvalues of selfadjoint operators on a Hilbert space depending nonlinearly on an eigenparameter usually assume differentiable dependence of the operator on the eigenparameter. In this paper we generalize these results to nonlinear problems which depend only continuously on the parameter. This result is applied to a class of variational eigenvalue problems which in particular contains the vibrations of plates with attached masses. Dedicated to Ivo Marek on the occasion of his 75th birthday 1. Introduction. In this paper we consider the nonlinear eigenvalue problem T(λ)x = 0 (1.1) where T(λ), λ J, is a selfadjoint and bounded operator on a Hilbert space H, and J is a real open interval which may be unbounded. As in the linear case T(λ) = λi A a parameter λ J is called an eigenvalue of problem (1.1) if the equation (1.1) has a nontrivial solution x 0. For a wide class of linear selfadjoint operators A : H H the eigenvalues of the linear eigenvalue problem (1.1) with T(λ) := λi A can be characterized by three fundamental variational principles, namely by Rayleigh s principle [13], by Poincaré s minmax characterization [12], and by the maxmin principle of Courant [3], Fischer [5] and Weyl [30]. These variational characterizations of eigenvalues are known to be very powerful tools when studying selfadjoint linear operators on a Hilbert space H. Bounds for eigenvalues, comparison theorems, interlacing results and monotonicity of eigenvalues can be proved easily with these characterizations, to name just a few. These variational principles were generalized to the nonlinear eigenvalue problem (1.1) where the Rayleigh quotient R(x) := Ax, x / x, x of a linear problem Ax = λx was replaced by the so called Rayleigh functional p, which is a homogeneous functional defined by the equation T(p(x))x, x = 0 for x 0. Notice that in the linear case T(λ) := λi A this is exactly the Rayleigh quotient. If the Rayleigh functional p is defined on the entire space H \ {0} then the eigenproblem (1.1) is called overdamped. This term is motivated by the quadratic eigenvalue problem T(λ)x = λ 2 Mx + λcx + Kx = 0 (1.2) governing the damped free vibrations of a system where M, C and K are symmetric and positive definite corresponding to the mass, the damping and the stiffness of the system, respectively. Institute of Numerical Simulation, Hamburg University of Technology, D Hamburg, Germany (voss@tu-harburg.de) 1

2 2 HEINRICH VOSS Assume that the damping C = α C depends on a parameter α 0. Then for α = 0 the system has purely imaginary eigenvalues corresponding to harmonic vibrations. Increasing α the eigenvalues move into the left half plane as conjugate complex pairs corresponding to damped vibrations. Finally they reach the negative real axis as double eigenvalues where they immediately split and move into opposite directions. When eventually all eigenvalues have become real, and all eigenvalues going to the right are right of all eigenvalues moving to the left the system is called overdamped. In this case the two solutions p ± (x) = ( α Cx, x ± α 2 Cx, x 2 4 Mx, x Kx, x )/(2 Mx, x ). of the quadratic equation T(λ)x, x = λ 2 Mx, x + λα Cx, x + Kx, x = 0 (1.3) are real, and they satisfy x 0 p (x) < inf x 0 p + (x). Hence, equation (1.3) defines two Rayleigh functionals p and p + corresponding to the intervals J := (, inf x 0 p + (x)) and J + := ( x 0 p (x), 0). For overdamped systems Hadeler [6], [7] generalized Rayleigh s principle proving that the eigenvectors are orthogonal with respect to the generalized scalar product (T(p(x)) T(p(y)))x, y, if p(x) p(y) [x, y] := p(x) p(y) T (p(x))x, y, if p(x) = p(y) which is symmetric, definite and homogeneous, but in general it is not bilinear. For finite dimensional overdamped problems Duffin [4] proved Poincaré s minmax characterization for quadratic eigenproblems and Rogers [14] for the general nonlinear eigenproblem. Infinite dimensional overdamped problems were treated by Hadeler [7], [8], Langer [10], Rogers [15], Turner [19], [20], and Werner [29], who proved generalizations of both characterizations, the minmax principle of Poincaré, and the maxmin principle of Courant, Fischer and Weyl. Barston [1] characterized a subset of the real eigenvalues of a nonoverdamped quadratic eigenproblem of finite dimension, and Werner and the author [26] studied the general nonoverdamped case and proved a minmax principle generalizing the characterization of Poincaré. The corresponding maxmin principle for nonoverdamped problems is contained in [21]. All papers mentioned above assume that the eigenproblem is differentiable with respect to λ. Without this assumption the minmax characterization was proved by Markus [11] and Hasanov [9] for overdamped problems. In this paper we generalize the approach in [26] to the nonoverdamped and non differentiable case. With the obtained characterizations we derive existence theorems and bounds for a class of variational eigenvalue problems generalizing corresponding results of Solov ëv [17]. A final section of this paper applies these results to a rational eigenproblem governing free vibrations of a plate with elastically attached loads. 2. A minmax principle. Let H be a Hilbert space with scalar product,, and let J R be an open interval which may be unbounded. We consider the nonlinear eigenvalue problem T(λ)x = 0 (2.1)

3 A MINMAX PRINCIPLE FOR NONLINEAR EIGENPROBLEMS 3 where T(λ) : H H, λ J, is a family of selfadjoint and bounded operators depending continuously on the parameter λ. As in the linear case T(λ) = λi A a parameter λ J is called an eigenvalue of problem (2.1) if the equation (2.1) has a nontrivial solution x 0, and x is called a corresponding eigenvector. We stress the fact that we are only concerned with real eigenvalues in J although T( ) may be defined on a larger subset of C, and T( ) may have additional eigenvalues in C \ J. To generalize the variational characterization of eigenvalues we need a generalization of the Rayleigh quotient. To this end we assume that (A 1 ) for every fixed x H, x 0 the real equation f(λ; x) := T(λ)x, x = 0 (2.2) has at most one solution λ =: p(x) J. Then f(λ; x) = 0 defines a functional p on some subset D(p) H which is called the Rayleigh functional of (2.1), and which is exactly the Rayleigh quotient in case of a linear eigenproblem T(λ) = λi A. Generalizing the definiteness requirement for linear pencils T(λ) = λb A we further assume that (A 2 ) for every x D(p) and every λ J with λ p(x) it holds that (λ p(x))f(λ, x) > 0. (2.3) The key to the variational principle in the nonoverdamped case is an appropriate enumeration of the eigenvalues. In general, the natural enumeration i.e. the first eigenvalue is the smallest one, followed by the second smallest one etc. is not reasonable. Instead, the number of an eigenvalue λ of the nonlinear problem (2.1) is inherited from the number of the eigenvalue 0 in the spectrum of the operator T(λ) based on the following consideration (cf. [26]). For j N and λ J let µ j (λ) := V S j T(λ)v, v min v V,v 0 v, v (2.4) where S j is the set of all j dimensional subspaces of H. We assume that (A 3 ) If µ n (λ) = 0 for some n N and some λ J, then for j = 1,...,n the remum in µ j (λ) is attained, and µ 1 (λ) µ 2 (λ) µ n (λ) are the n largest eigenvalues of the linear operator T(λ). Conversely, if µ = 0 is an eigenvalue of the operator T(λ), then µ n (λ) = 0 for some n N. Definition 2.1. λ J is an nth eigenvalue of T( ) if µ n (λ) = 0 for n N. Condition (A 3 ) is satisfied for example if for every λ J there exists ν(λ) > 0 such that T(λ) ν(λ)i is a compact operator since then the classical maxmin charactrization for linear selfadjoint operators applies. More general conditions are contained in [28]. Our main result is contained in the following theorem which generalizes the minmax characterization for nonoverdamped nonlinear eigenproblems in [26]. In that paper we assumed that T(λ) is differentiable in J H, and it holds that f(λ; x) λ > 0 for every x D(p). (2.5) λ=p(x) Our more general result requires some modifications of the proof given in [26].

4 4 HEINRICH VOSS Theorem 2.2. Assume that the conditions (A 1 ), (A 2 ) and (A 3 ) are satisfied. Then for every n N there exists at most one nth eigenvalue, and the following characterization holds: λ n = min V Sn V D(p) v V D(p) p(v). (2.6) In order to prove Theorem 2.2 we need the following two Lemmata the first of which follows immediately from the implicit function theorem in the differentiable case. Lemma 2.3. Under the conditions (A 1 ) and (A 2 ) the domain of definition D(p) of the Rayleigh functional p is an open set. Proof. We assume that D(p) is not an open set. Then there exists x D(p) and a sequence {x n } H \ D(p) with x n x, and the continuity of T(λ) yields f(λ; x n ) = T(λ)x n, x n T(λ)x, x = f(λ; x) for every λ J. For λ 1 < p(x) it follows from (A 2 ) that f(λ 1 ; x) < 0, and thus it follows for sufficiently large n N that f(λ 1 ; x n ) < 0. Analogously, we obtain f(λ 2 ; x n ) > 0 for λ 2 > p(x) and sufficiently large n N. Hence, for n large enough the continuous function f( ; x n ) has a root in (λ 1, λ 2 ) contradicting {x n } H \ D(p). Lemma 2.4. Let λ J, and assume that V is a finite dimensional subspace of H such that V D(p) Then it holds that λ Proof. Let < = > p(x) min x V ˆλ = T(λ)x, x < = > 0 (2.7) p(x) J. (2.8) Then there exists a sequence {x n } D(p) V with p(x n ) ˆλ and x n = 1. Without loss of generality we assume that {x n } converges to some ˆx (notice that the dimension of V is finite), and the continuity of T implies 0 = lim n T(p(x n))x n, x n = T(ˆλ)ˆx, ˆx. Hence, ˆx D(p) and p(ˆx) = ˆλ = max x D(p) V p(x), and it follows that If min T(ˆλ)x, x T(ˆλ)ˆx, ˆx = 0. x V min T(ˆλ)x, x = T(ˆλ)ŷ, ŷ < 0 x V and without loss of generality T(ˆλ)ˆx, ŷ 0, then it follows that T(ˆλ)(ˆx + tŷ), ˆx + tŷ = 2t T(ˆλ)ˆx, ŷ + t 2 T(ˆλ)ŷ, ŷ < 0 (2.9)

5 A MINMAX PRINCIPLE FOR NONLINEAR EIGENPROBLEMS 5 for every t > 0. Since V D(p) is an open set, ˆx + tŷ D(p) for sufficiently small t > 0, and (2.9) yields p(ˆx + tŷ) > ˆλ contradicting (2.8). Hence, we have shown: Conversely, if λ = p(x) min T(λ)x, x = 0. (2.10) x V f(λ, x) = T(λ)x, x min y V T(λ)y, y = 0 = T(λ)ȳ, ȳ, then ȳ D(p) with p(ȳ) = λ, and condition (A 2 ) yields p(x) λ for every x V D(p). Thus, we have λ = p(x) min T(λ)x, x = 0. (2.11) x V Let min T(λ)x, x > 0. From (A 2) we obtain λ p(x) > 0 for every x x V V D(p), i.e. λ x D(p) V p(x), and according to (2.11) it even holds that λ > p(x). Finally, let T(λ)ȳ, ȳ = min T(λ)x, x < 0. x V If λ > p(x), then λ > p( x) for some x V D(p), and therefore (T(λ) x, x) > 0. The continuous function g(t) = T(λ)((1 t) x + tȳ), (1 t) x + tȳ, has opposite signs at the end points of the interval [0, 1]. Hence, there exists a root t (0, 1), and with w = (1 t) x + tȳ V D(p) we obtain p(w) = λ, contradicting λ > p(x), which completes the proof. We are now in the position to prove Theorem 2.2. Proof. (of Theorem 2.2) Let λ n be an nth eigenvalue of problem (2.1). Then µ n (λ n ) = 0, and the remum in (2.4) is attained for some V n, i.e. min v V T(λ n )v, v = 0. For every V S n we have min v V T(λ n )v, v 0, and Lemma 2.4 implies and therefore it holds that p(x) λ n = p(x), x V 0 D(p) λ n = min V S n V D(p) v V D(p) p(v). The following two Theorems were proved in [26]. Their proofs do not require the differentiability of f, but only that D(p) is an open set and Lemma 2.4. Therefore they are also valid if assumption (2.5) is replaced by condition (A 2 ).

6 6 HEINRICH VOSS Theorem 2.5. Under the conditions (A 1 ), (A 2 ) and (A 3 ) let λ n := inf V Sn V D(p) v V D(p) p(v) J. (2.12) Then λ n is an nth eigenvalue of (2.1), and (2.6) holds. Theorem 2.6. Assume that conditions (A 1 ), (A 2 ) and (A 3 ) are satisfied, and that J contains a first eigenvalue λ 1 = inf x D(p) p(x). If λ n J for some n N then every V H j with V D(p) and λ j = u V D(p) p(u) is contained in D(p) {0}, and the characterization (2.6) can be replaced by λ j = min V H j max p(v) v V 1 j = 1,...,n. (2.13) V 1 D(p) {0} Finally, the proof of the maxmin characterization derived in [21] does not require the differentiability of f but only Lemma 2.3, and therefore it holds that Theorem 2.7. Assume that the conditions (A 1 ), (A 2 ) and (A 3 ) are satisfied. If there is an n-th eigenvalue λ n J of problem (2.1), then λ n = max V H n 1 V D(p) inf p(v), (2.14) v V D(p) and the maximum is attained by W := span{u 1,..., u n 1 } where u j denotes an eigenvector corresponding to the j-largest eigenvalue µ j (λ n ) of T(λ n ). 3. A variational eigenvalue problem. We consider the nonlinear eigenvalue problem in variational form Find λ R and u H, u 0, such that a(u, v, λ) = λb(u, v, λ) for every v H. (3.1) We assume that the following conditions are satisfied: (i) a : H H J R is an H-elliptic, continuous and symmetric bilinear form, i.e. there exist positive functions α 1, α 2 : J R + such that for λ J it holds that α 1 (λ) u 2 a(u, u, λ), and a(u, v, λ) α 2 (λ) u v for every u, v H. (ii) b : H H J R is a symmetric, completely continuous and positive definite bilinear form on H, i.e. there exists β 2 : J R such that 0, < b(u, v, λ) for every u H \ {0} and b(u, v) β 2 (λ) u v for every u, v H, and if {u n }, {v n } H are weakly convergent sequences such that u n u, v n v then b(u n, v n, λ) b(u, v, λ). (iii) With respect to the eigenparameter λ, we assume that a and b are uniformly continuous, i.e. lim α(λ, µ) = 0 and lim β(λ, µ) = 0 µ λ µ λ

7 where A MINMAX PRINCIPLE FOR NONLINEAR EIGENPROBLEMS 7 a(u, v, λ) a(u, v, µ) α(λ, µ) u v, b(u, v, λ) b(u, v, µ) β(λ, µ) u v. It is well known (cf. Weinberger [27]) that for fixed λ J there exist a bounded operator A(λ) : H H and a completely continuous operator B(λ) : H H such that a(u, v, λ) = A(λ)u, v and b(u, v, λ) = B(λ)u, v for every u, v H. Hence, the variational eigenvalue problem (3.1) is equivalent to problem (2.1) with T(λ) = λb(λ) A(λ), (3.2) and it follows from (iii) that A and B (and therefore T) depend continuously on λ. From the maxmin characterization of Poincaré for linear eigenproblems it follows that condition (A 3 ) is satisfied. For fixed u H f(λ; u) = λb(u, u, λ) a(u, u, λ), (3.3) and the general conditions in Section 2 are fulfilled, if for fixed u H the real equation f(λ; u) = 0 has at most one solution p(u) in J, and if condition (2.3) holds. This is, for example, the case if (iv) for fixed u H \ {0} the Rayleigh quotient R(λ; u) = a(u, u, λ) b(u, u, λ) (3.4) of the linear eigenproblem a(u, v, λ) = µb(u, v, λ) is monotonically not increasing with respect to λ J. Under the conditions (i), (ii), (iii), and (iv) the linear eigenvalue problem: find u 0 and γ such that a(u, v, µ) = γ(µ)b(u, v, µ) for every v H has an infinite number of eigenvalues 0 < γ 1 (µ) γ 2 (µ)... which depend continuously on µ and which are monotonically decreasing with respect to µ. Obviously µ is an eigenvalue of the nonlinear eigenvalue problem (3.1) if and only if µ = γ j (µ) for some j. This was the basis of existence results for (3.1) proved by Solov ëv [17, 18]. Our results for the nonlinear eigenproblem (3.1) are based on the following lemma: Lemma 3.1. Assume that (i), (ii), (iii) and (iv) are satisfied, and let u H\{0} with R(κ, u) J for some κ J. Then u D(p) and the following inequalities hold min(κ, R(κ, u)) p(u) max(κ, R(κ, u)). (3.5) Proof. Condition (iv) yields f(r(κ, u), u) b(u, u, R(κ, u)) a(u, u, R(κ, u)) = R(κ, u) b(u, u, R(κ, u)) { = R(κ, u) R(R(κ, u), u) } 0 for { R(κ, u) κ R(κ, u) κ }

8 8 HEINRICH VOSS and similarly f(κ, u) b(u, u, κ) = κ R(κ, u){ } 0 for { R(κ, u) κ R(κ, u) κ }. In either case, R(κ, u) κ and R(κ, u) κ, f(, u) changes its sign between R(κ, u) and κ, and therefore u D(p) und the inclusion (3.5) holds. We first consider the case that (3.1) has a first eigenvalue in J. Theorem 3.2. Assume that (i), (ii), (iii)and (iv) hold, and let κ J. Suppose that there exists η J such that η min u H\{0} R(η, u) 0, (3.6) and that the linear eigenvalue problem: find µ R and u H \ {0} with a(u, v, κ) = µb(u, v, κ) for every v H (3.7) has r eigenvalues µ 1 µ 2 µ r in J. Then the nonlinear eigenvalue problem (3.1) has r eigenvalues λ 1 λ 2... λ r in J, and the following inclusion holds min(µ j, κ) λ j max(µ j, κ), j = 1,...,r. (3.8) Proof. We first proof the existence of r eigenvalues of (3.1). From (3.6) it follows that inf u D(p) p(u) η. Hence, by Theorem 2.6 we only have to show that there exist a subspace W of dimension r with W D(p) and u W D(p) p(u) J. Let W be the invariant subspace of problem (3.7) which is spanned by the eigenvectors corresponding to µ 1,..., µ r. Then it holds that µ r = min dim V =r max v V, v 0 R(κ, v) = max R(κ, v) J. v W, v 0 Hence, η R(κ, v) µ r for every v W \ {0}, and it follows from Lemma 3.1 that W \ {0} D(p), and p(v) max(κ, R(κ, v)) max(κ, µ r ). By Theorem 2.6 problem (3.1) has at least r eigenvalues λ 1 λ r in J. We now prove inequality (3.8). For j {1,..., r} let Z j be the j dimensional subspace with µ j = min dim V =j max v V, v 0 R(κ, v) = max Then it holds that Z W D(p) {0}, and from v Z, z 0 R(κ, v). p(v) max(κ, R(κ, v)) for every v Z, z 0

9 A MINMAX PRINCIPLE FOR NONLINEAR EIGENPROBLEMS 9 we obtain λ j = min dim V = j V D(p) {0} max p(v) max p(v) v V, v 0 v Z, v 0 max(κ, max v Z, v 0 R(κ, v)) = max(κ, µ j), which proves the upper bound of λ j in (3.8). Let Y D(p) {0}, dimy = j with λ j = min dim V = j V D(p) {0} max p(v) = max p(v). v V, v 0 v Y, v 0 For u Y, u 0 with p(u) κ we obtain p(u) = a(u, u, p(u)) a(u, u, κ) = R(κ, u). b(u, u, p(u)) b(u, u, κ) Hence, if p(u) κ for every u Y, u 0, then λ j = max p(u) u Y, u =0 max u Y, u =0 R(κ, u) min dim V =j max R(κ, u) = µ j, u V, u =0 and if there exists u Y, u 0 with p(u) > κ, then it follows λ j = max v Y, v 0 p(v) p(u) > κ. In either case we have λ j min(κ, µ j ), which completes the proof. Next we consider the case that the interval J does not contain a first eigenvalue. We first prove the following result Theorem 3.3. Assume that the conditions (i), (ii), (iii), and (iv) are satisfied, and let κ J. Suppose that κ J and J contains an mth eigenvalue µ m of the linear eigenproblem (3.7), then the nonlinear eigenvalue problem (3.1) has an mth eigenvalue λ m J, and it holds that min(µ m, κ) λ m max(µ m, κ). (3.9) Proof. We show that (α) there exists a subspace W, dimw = m with W D(p) and (β) u W D(p) p(u) max(κ, µ m ) (3.10) p(u) min(κ, µ m ) for every V with dimv = m and V D(p). u V D(p) Then it holds that λ m := inf dim V = m V D(p) u V D(p) p(u) J,

10 10 HEINRICH VOSS and the inclusion (3.9) follows. (α): Let dimw = m, and let w W with µ m = max u W, u =0 R(κ, u) = R(κ, w). Then Lemma 3.1 yields w D(p), and W D(p). R(κ, u) µ m for every u W, u 0 implies a(u, u, κ) b(u, u, κ) µ m for every u W, u 0. If κ µ m, it follows immediately f(κ, u) 0, and if µ m κ, then we obtain f(µ m, u) 0 from the monotonicity of R(λ, u). In either case it holds that f(σ, u) 0 for σ = max(κ, µ m ) for every u W, u 0 and condition (ii) yields inequality (3.10). (β) is proved by contradiction: Assume that there exists V, dimv = m with V D(p) and Let u V V, u V 0 with u V D(p) ρ := R(κ, u V ) = p(u) < min(κ, µ m ). (3.11) max u V, u =0 R(κ, u). Then ρ J, since otherwise by Lemma 3.1 it would follow that u V p(u V ) min(κ, R(κ, u V )), i.e. D(p) and u V D(p) Let σ := min(κ, µ m ). Then σ ρ implies p(u) p(u V ) min(κ, R(κ, u V )) min(κ, µ m ). f(σ, u V ) b(u V, u V, σ) = σ a(u V, u V, σ) b(u V, u V, σ) σ ρ 0, i.e. f(σ, u V ) 0. For fixed u V D(p) let w(t) := tu + (1 t)u V and φ(t) = f(σ, w(t)). Then φ is continuous on [0, 1], and it holds that f(σ, u V ) = φ(0) 0 f(σ, u) = φ(1). Hence, there exists ˆt [0, 1] with f(σ, w(ˆt)) = 0, i.e. w(ˆt) V D(p) and p(w(ˆt)) = min(κ, µ m ) contradicting (3.11). From Theorem 3.2 we immediately obtain Theorem 3.4. Assume that (i), (ii), (iii), and (iv) are satisfied. Let κ 1, κ 2 J with κ 1 < κ 2, and pose that for κ = κ j, j = 1, 2 the linear eigenvalue problem (3.7) has exactly m j eigenvalues which do not exceed κ j Then the nonlinear eigenvalue problem (3.1) has at least m 2 m 1 eigenvalues in J. In (κ 1, κ 2 ] there are exactly m 2 m 1 eigenvalue λ m1+1 λ m2, where λ j is a jth eigenvalue, and for every parameter κ [κ 1, κ 2 ] it holds that min(κ, µ j (κ)) λ j max(κ, µ j (κ)), für j = m 1 + 1,...,m 2. (3.12) Here µ j (κ) denotes the j smallest eigenvalue of (3.7) corresponding to κ. Proof. Follows immediately from Theorems 3.3 und 2.6.

11 A MINMAX PRINCIPLE FOR NONLINEAR EIGENPROBLEMS Vibration of plates with masses. The vertical deflection w(x, t) of a thin isotropic clamped plate with elastically attached loads is governed by the equations [16, 24] q Lw(x, t) + ρh 2 t 2 w(x, t) + d 2 m j dt 2 ξ j(t)δ xj w = 0, x Ω, t > 0 (4.1) j=1 w(x, t) = n w(x, t) = 0, x Ω, t > 0 (4.2) m j d 2 dt 2 ξ j + k j (ξ j (t) w(x j, t)) = 0, t > 0, j = 1,...,q. (4.3) Here Ω R 2 is a bounded Lipschitz domain which is occupied by the plate, ρ is the mass per volume density, and h the thickness of the plate. L = D 2 is the plate operator, D = Eh 3 /12(1 ν 2 ) is the flexural regidity of the plate, E is the Young modulus and ν is the Poisson number. For j = 1,...,q at x j Ω a load m j is joined elastically to the plate with stiffness coefficient k j, and ξ j denotes the displacement of the mass m j. Using the ansatz w(x, t) = u(x)e iωt and ξ j (t) = c j e iωt characterizing the eigenmodes and eigenfrequencies of the vibrating plate, and eliminating c j we obtain the rational eigenproblem Lu(x) = λρhu(x) + q j=1 λσ j σ j λ m jδ xj u, x Ω (4.4) u(x) = n u(x) = 0, x Ω (4.5) where λ = ω 2 and σ j = k j /m j. Let H = H0 2 (Ω) denote the Sobolev space with the usual scalar product. Multiplying (4.4) by v H and integrating by parts one obtains the variational form of problem (4.4), (4.5) ( p σ ) j A(u, v) = λ B(u, v) + σ j λ C j(u, v) (4.6) where Ω j=1 A(u, v) = D u v dx, (4.7) B(u, v) = ρhuv dx, (4.8) Ω C j (u, v) = m j u(x j )v(x j ). (4.9) For J = (0, min j σ j ) the conditions of Theorem 3.2 are obviously satisfied with a(u, v, λ) = A(u, v) and b(u, v, λ) = B(u, v) + p j=1 σ j σ j λ C j(u, v). Hence, if for some κ min j σ j the linear eigenproblem: find u H0 2 (Ω) and µ R such that ( p σ ) j A(u, v) = µ B(u, v) + σ j κ C j(u, v) for every v H0(Ω) 2 (4.10) j=1

12 12 HEINRICH VOSS has r eigenvalues 0 < µ 1 µ 2 µ r < min j σ j, then the nonlinear eigenproblem (4.4), (4.5) has r eigenvalues λ j, and min(µ j, κ) λ j max(µ j, κ). Since the Rayleigh quotient of problem (4.10) is less than the one of the pure plate problem without masses: find u 0 and λ such that A(u, v) = λb(u, v) for every v H 2 0(Ω), (4.11) it follows immediately that the nonlinear eigenproblem (4.4), (4.5) has at least as many eigenvalues in J as (4.11). If J = (ξ, η) R + is an interval which does not contain a pole σ j then the conditions of Theorem 3.4 are satisfied with a(u, v, λ) = A(u, v) + σ j ξ b(u, v, λ) = B(u, v) + σ j η λσ j λ σ j C j (u, v), (4.12) σ j σ j λ C j(u, v), (4.13) and the number of eigenvalues of (4.4), (4.5) and enclosures of the eigenvalues can be obtained from linear eigenproblems. In particular, the interval J = (max j σ j, ) contains an infinite number of eigenvalues. In a similar fashion one gets existence results and minmax characterizations of eigenvalues of rational eigenvalue problems governing free vibrations of fluid-solid structures as considered in Conca, Planchard, Vanninathan [2, 21, 22, 25]. Numerical Example Consider the clamped plate occupying the domain Ω = (0, 4) (0, 3) with constant coefficients ρ = d = 1. We assume that 6 masses are attached to the plate at x 1 = (1, 1), x 2 = (2, 1), x 3 = (3, 1), x 4 = (1, 2), x 5 = (2, 2) and x 6 = (3, 2), where σ 1 = σ 2 = σ 3 = 1000, σ 4 = σ 6 = 2000, and σ 5 = 3000, and m 1 = m 2 = m 3 = 1, m 4 = m 6 = 1/2 and m 5 = 1/3. We discretized the eigenproblem by Bogner-Fox-Schmit elements on a quadratic mesh with stepsize h = 0.05 which yielded a matrix eigenvalue problem Kx = λmx λ 1000 λ C 1x λ 2000 λ C 2x λ 3000 λ C 3x (4.14) of dimension Here K and M are the stiffness and the mass matrix of the plate without masses, and C j for j = 1, 2, 3 is a diagonal matrix of rank 3, 2 and 1, respectively, corresponding to the loads {m 1, m 2, m 3 }, {m 4, m 6 } and m 5. For κ := σ 1 ε, ε > 0 small enough the linear eigenvalue problem ( Kx = λ Mx κ C 1x κ C 2x ) 3000 κ C 3 x (4.15) has 24 eigenvalues which are less than σ ε, and by Theorem 3.2 problem (4.14) has 24 eigenvalues smaller than the smallest pole σ 1 enumerated λ 1,...,λ 24. By the minmax characterization the eigenvalues of (4.14) are upper bounds of the corresponding eigenvalues of (4.10), and therefore (4.10) has at least 24 eigenvalues not exceeding σ 1. In a similar manner one gets from Theorem 3.4 with (3.7) (where a and b are discrete versions of (4.12) and (4.13)) that the interval J 2 = (σ 1, σ 2 ) contains 8 eigenvalues of (4.14) enumerated λ 22,..., λ 29, and the interval J 3 contains 9 eigenvalues

13 A MINMAX PRINCIPLE FOR NONLINEAR EIGENPROBLEMS 13 enumerated ˆλ 29,..., ˆλ 36. Finally, there are eigenvalues greater than σ 3 the smallest of which being a 36th one. This example demonstrates that the enumeration of eigenvalues in (A 3 ) corresponds to the natural ordering only if inf x D(p) p(x) is contained in the corresponding interval, and that there may exist eigenvalues which carry the same number but are different from each other (λ 22 λ 22, e.g.). Notice that eigenvalues in (0, σ 1 ), (σ 1, σ 2 ), (σ 2, σ 3 ), and (σ 3, ) can be determined safely by the nonlinear Arnoldi method [23, 24]. Acknowledgement. This paper was finished while the author was visiting the National Center for Theoretical Sciences, Mathematics Division, in Hsinchu, Taiwan; he thanks the Center for its hospitality. REFERENCES [1] E.M. Barston. A minimax principle for nonoverdamped systems. Int. J. Engrg. Sci., 12: , [2] C. Conca, J. Planchard, and M. Vanninathan. Fluid and Periodic Structures, volume 24 of Research in Applied Mathematics. Masson, Paris, [3] R. Courant. Über die Eigenwerte bei den Differentialgleichungen der mathematischen Physik. Math. Z., 7:1 57, [4] R.J. Duffin. A minimax theory for overdamped networks. J. Rat. Mech. Anal., 4: , [5] E. Fischer. Über quadratische Formen mit reellen Koeffizienten. Monatshefte für Math. und Phys., 16: , [6] K. P. Hadeler. Mehrparametrige und nichtlineare Eigenwertaufgaben. Arch. Ration. Mech. Anal., 27: , [7] K. P. Hadeler. Variationsprinzipien bei nichtlinearen Eigenwertaufgaben. Arch. Ration. Mech. Anal., 30: , [8] K. P. Hadeler. Nonlinear eigenvalue problems. In R. Ansorge, L. Collatz, G. Hämmerlin, and W. Törnig, editors, Numerische Behandlung von Differentialgleichungen, ISNM 27, pages Birkhäuser, Stuttgart, [9] M. Hasanov. An approximation method in the variational theory of the spectrum of operator pencils. Acta Applicandae Mathematicae, 71: , [10] H. Langer. Über stark gedämpfte Scharen im Hilbertraum. J. Math. Mech., 17: , [11] A.S. Markus. Introduction to the Spectral Theory of Polynomial Operator Pencils, volume 71 of, Translations of Mathematical Monographs. AMS, Providence, [12] H. Poincaré. Sur les equations aux dérivées partielles de la physique mathématique. Amer. J. Math., 12: , [13] Rayleigh. Some general theorems relating to vibrations. Proc. London Math. Soc., 4: , [14] E.H. Rogers. A minimax theory for overdamped systems. Arch. Ration. Mech. Anal., 16:89 96, [15] E.H. Rogers. Variational properties of nonlinear spectra. J. Math. Mech., 18: , [16] S. I. Solov ëv. Eigenvibrations of a plate with elastically attached loads. Preprint SFB393/03-06, Sonderforschungsbereich 393 an der Technischen Universitt Chemnitz, Technische Universität, D Chemnitz, Germany, Available at [17] S.I. Solov ëv. The finite-element method for symmetric nonlinear eigenvalue problems. Comput. Math. Math. Phys., 37: , [18] S.I. Solov ëv. Preconditioned iterative methods for a class of nonlinear eigenvalue problems. Linear Algebra Appl., 415: , [19] R.E.L. Turner. Some variational principles for nonlinear eigenvalue problems. J. Math. Anal. Appl., 17: , [20] R.E.L. Turner. A class of nonlinear eigenvalue problems. J. Func. Anal., 7: , [21] H. Voss. A maxmin principle for nonlinear eigenvalue problems with application to a rational spectral problem in fluid solid vibration. Applications of Mathematics, 48: , [22] H. Voss. A rational spectral problem in fluid solid vibration. Electronic Transactions on Numerical Analysis, 16:94 106, 2003.

14 14 HEINRICH VOSS [23] H. Voss. An Arnoldi method for nonlinear eigenvalue problems. BIT Numerical Mathematics, 44: , [24] H. Voss. Eigenvibrations of a plate with elastically attached loads. In P. Neittaanmäki, T. Rossi, S. Korotov, E. Onate, J. Periaux, and D. Knörzer, editors, Proceedings of the European Congress on Computational Methods in Applied Sciences and Engineering. ECCOMAS 2004, Jyväskylä, Finland, ISBN , available at [25] H. Voss. Locating real eigenvalues of a spectral problem in fluid-solid type structures. J. Appl. Math., 2005:37 48, [26] H. Voss and B. Werner. A minimax principle for nonlinear eigenvalue problems with applications to nonoverdamped systems. Math. Meth. Appl. Sci., 4: , [27] H.F. Weinberger. Variational Methods for Eigenvalue Approximation, volume 15 of Regional Conference Series in Applied Mathematics. SIAM, Philadelphia, [28] A. Weinstein and W.F. Stenger. Methods of Intermediate Problems for Eigenvalues. Academic Press, New York, [29] B. Werner. Das Spektrum von Operatorenscharen mit verallgemeinerten Rayleighquotienten. Arch. Ration. Mech. Anal., 42: , [30] H. Weyl. Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung). Math.Ann., 71: , 1912.