비만아 부모의 돌봄 경험: q 방법론

Can one hear the shape of a drum

TOTAL MEAN CURVATURE AND SUBMANIFOLDS OF FINITE TYPE (World Scientific Series in Pure Mathematics—Volume 1) By Bang-Yen Chen: pp. 352. £15.70. (Published by World Scientific Publishing, distributed by John Wiley & Sons Ltd, 1984.)

We summarize the properties of eigenvalues and eigenfunctions of the Laplace operator in bounded Euclidean domains with Dirichlet, Neumann or Robin boundary condition. We keep the presentation at a level accessible to scientists from various disciplines ranging from mathematics to physics and computer sciences. The main focus is put onto multiple intricate relations between the shape of a domain and the geometrical structure of eigenfunctions.

Geometrical structure of Laplacian eigenfunctions

In this paper, we investigate an eigenvalue problem of Dirichlet Laplacian on a bounded domain Ω in an n-dimensional Euclidean space Rn. If λk+1 is the (k + 1)th eigenvalue of Dirichlet Laplacian on Ω, then, we prove that, for n ≥ 41 and \(k\geq 41, \lambda_{k+1}\leq k^{\frac2n}\lambda_1\) and, for any n and \(k, \lambda_{k+1}\leq C_{0}(n,k) k^{\frac2n}\lambda_1\) with \(C_0(n,k)\leq {j^{2}_{n/2,1}}/{j^{2}_{n/2-1,1}}\), where jp,k denotes the k-th positive zero of the standard Bessel function Jp(x) of the first kind of order p. From the asymptotic formula of Weyl and the partial solution of the conjecture of Polya, we know that our estimates are optimal in the sense of order of k.

Bounds on eigenvalues of Dirichlet Laplacian

In this paper, we study eigenvalues of Laplacian on either a bounded connected domain in an n-dimensional unit sphere Sn(1), or a compact homogeneous Riemannian manifold, or an n-dimensional compact minimal submanifold in an N-dimensional unit sphere SN(1). We estimate the k+1-th eigenvalue by the first k eigenvalues. As a corollary, we obtain an estimate of difference between consecutive eigenvlaues. Our results are sharper than ones of P. C. Yang and Yau [25], Leung [19], Li [20] and Harrel II and Stubbe [12], respectively. From Weyl’s asymptotical formula, we know that our estimates are optimal in the sense of the order of k for eigenvalues of Laplacian on a bounded connected domain in an n-dimensional unit sphere Sn(1).

Estimates on Eigenvalues of Laplacian

Let D be a connected bounded domain in an n-dimensional Euclidean space R n . Assume that 0 < λ 1 < λ 2 ≤ ··· ≤λ k ≤··· are eigenvalues of a clamped plate problem or an eigenvalue problem for the Dirichlet biharmonic operator: Then, we give an upper bound of the (k+1)-th eigenvalue λ k+1 in terms of the first k eigenvalues, which is independent of the domain D, that is, we prove the following: Further, a more explicit inequality of eigenvalues is also obtained.

/pdf/inequalities-for-eigenvalues-of-a-clamped-plate-problem-4lfdtkul26.pdf

Inequalities for eigenvalues of a clamped plate problem

For a bounded domain with a piecewise smooth boundary in an n-dimensional Euclidean space R n , we study eigenvalues of the Dirichlet eigenvalue problem of the Laplacian. By making use of a fact that eigenfunctions form an orthonormal basis of L 2 () in place of the Rayleigh-Ritz formula, we obtain inequalities for eigenvalues of the Laplacian. In particular, we study a conjecture of Payne, Polya and Weinberger on lower order eigenvalues. Our results will become an important progress for solving this conjecture.

Inequalities for eigenvalues of the laplacian

For a compact spin manifold M isometrically embedded into Euclidean space, we derive the extrinsic estimates from above and below for eigenvalues of the square of the Dirac operator, which depend on the second fundamental form of the embedding. We also show the bounds of the ratio of the eigenvalues. Since the unit sphere and the projective spaces admit the standard embedding into Euclidean spaces, we also obtain the corresponding results for their compact spin submanifolds.

/pdf/extrinsic-estimates-for-eigenvalues-of-the-laplace-operator-48338br4zn.pdf

Qing-Ming Cheng

Papers

Bounds on eigenvalues of Dirichlet Laplacian

Estimates on Eigenvalues of Laplacian

Inequalities for eigenvalues of a clamped plate problem

Inequalities for eigenvalues of the laplacian

Extrinsic estimates for eigenvalues of the Laplace operator