Universal bounds for eigenvalues of the biharmonic operator

作者：Wang Qiaoling; Xia Changyu^*

来源：Journal of Mathematical Analysis and Applications, 2010, 364(1): 1-17.

DOI：10.1016/j.jmaa.2009.11.014

• Universal bounds
• Eigenvalues
• Biharmonic operator
• Sphere
• Euclidean space
• Minimal submanifolds

In this paper, we study the eigenvalues of the clamped plate problem:
{(Delta 2u = lambda u, in D,) (u vertical bar partial derivative D = partial derivative u/partial derivative v vertical bar partial derivative D = 0,)
where D is a bounded connected domain in an n-dimensional complete minimal submanifold of a unit m-sphere S(m)(1) or of an m-dimensional Euclidean space R(m). Let 0 < lambda(1) < lambda(2) <= ... <= lambda(k) <= ... be the eigenvalues of the above problem. We obtain universal bounds on lambda(k+1) in terms the first k eigenvalues independent of the domains. For example, when D is contained in an n-dimensional complete minimal submanifold of S(m)(1), we show that
lambda(k+1) - 1/k Sigma(k)(i=1) lambda(i) <= i/kn {Sigma(k)(i=1) (lambda(k+1) - lambda(i))(1/2) ((2n+4)lambda(1/2)(i) + n(2)}(1/2) . {Sigma(k)(i=1)(lambda(k+1) - lambda(i))(1/2) (4 lambda(1/2)(i) + n(2))}(1/2),
from which one can obtain a more explicit upper bound on lambda(k+1) in terms of lambda 1, ..., lambda(k) (see Corollary 1). When D is contained in a complete n-dimensional minimal submanifold of R(m), we prove the inequality
lambda(k+1) <= 1/k Sigma(k)(i=1)lambda(k) + (8(n+2)/n(2))(1/2) 1/k Sigma(k)(i=1) (lambda(i)(lambda(k+1) - lambda(i)))(1/2)
which generalizes the main theorem in Cheng and Yang (2006) [10] that states that the same estimate holds when D is a connected and bounded domain in R(m).

• 出版日期2010-4-1