Zc. Chen et Cl. Qian, ON THE UPPER BOUND OF EIGENVALUES FOR ELLIPTIC-EQUATIONS WITH HIGHER ORDERS, Journal of mathematical analysis and applications, 186(3), 1994, pp. 821-834
Let Omega be a bounded domain in R(m) with piecewise smooth boundary.
We consider the upper bound of the (n + 1)th eigenvalue lambda(n + 1)
for the two problems [GRAPHICS] and [GRAPHICS] where l and r are posit
ive integers with l>r, v is the unit outward normal to partial derivat
ive Omega and P(t)=a(l-r)t(l)+a(l-r-1)t(l-1)+...+a(1)t(r+1) with the c
onstant coefficients a(l-r)=1, a(i) greater than or equal to 0 for i=1
, 2,..., l-r-1. The bounds of lambda(n+1) are expressed in terms of th
e preceding eigenvalues. This generalizes the inequalities obtained by
Payne, Polya, Weinberger, Protter, Hile, and Yeh. (C) 1994 Academic P
ress, Inc.