Universal inequalities for eigenvalues of a clamped plate problem on a hyperbolic space

Abstract:

In this paper, we investigate universal inequalities for eigenvalues of a clamped plate problem on a bounded domain in an $n$-dimensional hyperbolic space. It is well known that, for a bounded domain in the $n$-dimensional Euclidean space, Payne, Pólya and Weinberger (1955), Hook (1990) and Chen and Qian (1990) studied universal inequalities for eigenvalues of the clamped plate problem. Recently, Cheng and Yang (2006) have derived the Yang-type universal inequality for eigenvalues of the clamped plate problem on a bounded domain in the $n$-dimensional Euclidean space, which is sharper than the other ones. For a domain in a unit sphere, Wang and Xia (2007) have also given a universal inequality for eigenvalues. For a bounded domain in the $n$-dimensional hyperbolic space, although many mathematicians want to obtain a universal inequality for eigenvalues of the clamped plate problem, there are no results on universal inequalities for eigenvalues. The main reason that one could not derive a universal inequality is that one cannot find appropriate trial functions. In this paper, by constructing “nice” trial functions, we obtain a universal inequality for eigenvalues of the clamped plate problem on a bounded domain in the hyperbolic space. Furthermore, we can prove that if the first eigenvalue of the clamped plate problem tends to $\tfrac {(n-1)^4}{16}$ when the domain tends to the hyperbolic space, then all of the eigenvalues tend to $\tfrac {(n-1)^4}{16}$.