Can one hear the shape of a drum

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

Spectral theory for perturbed Krein Laplacians in nonsmooth domains

Universal bounds for eigenvalues of the biharmonic operator on Riemannian manifolds

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.

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Inequalities for eigenvalues of a clamped plate problem

In this paper, we study n-dimensional complete minimal hypersurfaces in a unit sphere. We prove that an n-dimensional complete minimal hypersurface with constant scalar curvature in a unit sphere with f3 constant is isometric to the totally geodesic sphere or the Clifford torus if S ≤ 1.8252n−0.712898, where S denotes the squared norm of the second fundamental form of this hypersurface.

Chern's conjecture on minimal hypersurfaces

It is well known that the spectrum of Laplacian on a compact Riemannian manifold M is an important analytic invariant and has important geometric meanings. There are many mathematicians to investigate properties of the spectrum of Laplacian and to estimate the spectrum in term of the other geometric quantities of M . When M is a bounded domain in Euclidean spaces, a compact homogeneous Riemannian manifold, a bounded domain in the standard unit sphere or a compact minimal submanifold in the standard unit sphere, the estimates of the k + 1 -th eigenvalue were given by the first k eigenvalues (see [9], [12], [19], [20], [22], [23], [24] and [25]). In this paper, we shall consider the eigenvalue problem of the Laplacian on compact Riemannian manifolds. First of all, we shall give a general inequality of eigenvalues. As its applications, we study the eigenvalue problem of the Laplacian on a bounded domain in the standard complex projective space C P n ( 4 ) and on a compact complex hypersurface without boundary in C P n ( 4 ) . We shall give an explicit estimate of the k + 1 -th eigenvalue of Laplacian on such objects by its first k eigenvalues.

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Hongcang Yang

Papers

Bounds on eigenvalues of Dirichlet Laplacian

Estimates on Eigenvalues of Laplacian

Inequalities for eigenvalues of a clamped plate problem

Chern's conjecture on minimal hypersurfaces

Inequalities for eigenvalues of Laplacian on domains and compact complex hypersurfaces in complex projective spaces