ON THE UPPER BOUND OF EIGENVALUES FOR ELLIPTIC-EQUATIONS WITH HIGHER ORDERS

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.