We find a lower bound for the ratio between the first eigenvalue of any homogeneous thin plate G, which is clamped on its boundary, and the first eigenvalue of the spherical clamped plate having the same measure as G. In two dimensions, our bound is about 0.98.

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On the first eigenvalue of the clamped plate

We solve the biharmonic eigenvalue problem $\Delta^2u = \lambda u$
 and the buckling plate problem ${\Delta}^2u = - {\lambda}\Delta u$
 on the unit square using a highly accurate spectral Legendre--Galerkin method. We study the nodal lines for the first eigenfunction near a corner for the two problems. Five sign changes are computed and the results show that the eigenfunction exhibits a self similar pattern as one approaches the corner. The amplitudes of the extremal values and the coordinates of their location as measured from the corner are reduced by constant factors. These results are compared with the known asymptotic expansion of the solution near a corner. This comparison shows that the asymptotic expansion is highly accurate already from the first sign change as we have complete agreement between the numerical and the analytical results. Thus, we have an accurate description of the eigenfunction in the entire domain.

High precision solutions of two fourth order eigenvalue problems

We present a numerical study of the spectral gap of the Dirichlet Laplacian, γ(K) = λ2(K) − λ1(K), of a planar convex region K. Besides providing supporting numerical evidence for the long-standing gap conjecture that γ(K) ≥ 3π2/d2(K), where d(K) denotes the diameter of K, our study suggests new types of bounds and several conjectures regarding the dependence of the gap not only on the diameter, but also on the perimeter and the area. One of these conjectures is a stronger version of the gap conjecture mentioned above. A similar study is carried out for the quotient of the first two Dirichlet eigenvalues.

A numerical study of the spectral gap

We explain a numerical procedure to compute error bounds in H 2 0 () for eigenfunctions of elliptic eigenvalue problems of fourth order. Therefore, we compute a nite-element approximation and an upper bound for the defect in H ?2 (). Then, a theorem of Kato, eigenvalue inclusions and explicit embedding constants yield a pointwise error bound for the approximation. In order to control rounding errors, we use interval arithmetic. As an application, we prove the existence of nodal lines for the rst eigenfunction of the clamped plate and for the buckling plate in a square. AMS Symbol classiication: 65N25 The numerical computations in Bauer-Reiss 2] and Hackbusch-Hoffmann 12] strongly indicate the existence of nodal lines for the rst eigenfunction of the clamped plate 2 u ? u = 0; u 2 H 2 0 (); = (0; 1) 2 : Here, we present a numerical method to verify this result by pointwise error bounds for approximations of eigenfunctions. For the computation of error bounds for eigenfunctions we use a theorem of Kato (see sec. 2): the error of an approximate eigenfunction of some isolated eigenvalue can be bounded by the defect divided by the distance to the rest of the spectrum. For the application of the theorem we need an upper bound for the defect in H ?2 (). Due to the fact that 2 maps H 2 0 () isometrically onto H ?2 (), we get an error estimate in H 2 0 (). Now, explicit embeddings H 2 0 () ?! C 0 () resp. H 2 0 () ?! C 0;; () yield the result. Our approximations consist of piecewise polynomials and all norms can be evaluated using interval arithmetic within small error bounds. Thus we get guaranteed pointwise bounds. In some cases the existence of nodal lines can be proved by analytic considerations (cf. Coffman 5], 6], 7], 8], 9]); our method can be applied to more general domains and eigenvalue problems. Here, we also prove the existence of nodal lines for the rst eigenfunction of the buckling plate 2 u + u = 0; u 2 H 2 0 (); = (0; 1) 2 :

A numerical existence proof of nodal lines for the first eigenfunction of the plate equation

The wave method of Kelley and Rubinow [Ann. Physics, 9 (1960), p. 24–75] is extended to the biharmonic eigenvalue problem with rectangular or circular geometry and clamped boundary conditions. First, it is noted from the clues of computer graphics that mode shapes of a clamped circular plate and those of a circular membrane look very similar to each other. This suggests that plate and membrane should have very similar vibration behavior and leads to the assumption that the covering space of a rectangular plate is still a torus By adding several waves on the boundary, approximate eigenfrequency equations are derived. Their solutions are shown to agree remarkably with numerical solutions obtained by the Legendre-tau spectral method here and by the finite-element method elsewhere at all frequency ranges. The same idea is also applied to the circular plate and yields excellent agreement between the exact values of eigenfrequencies and the asymptotic solutions.

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Analysis of vibration eigenfrequencies of a thin plate by the Keller-Rubnow wave method I: clamped boundary conditions with rectangular or circular geometry

A recently published paper describes a numerical method for the fast solution of discretized elliptic eigenvalue problems. Here we report the results of this technique applied to the biharmonic (plate) equation. 46 eigenvalues (24 single and 11 double ones) are computed and the corresponding eigenfunctions are plotted.

Götz Hofmann

Papers

Results of the eigenvalue problem for the plate equation