On the Upper Bound of Eigenvalues for Elliptic Equations with Higher Orders

Let Ω be a bounded domain in Rm with piecewise smooth boundary. We consider the upper bound of the (n+1)th eigenvalue λn+1 for the two problems [formula] and [formula] where l and r are positive integers with l>r, v is the unit outward normal to ∂Ω, and P(t)=al−rtl+al−r−1tl−1+ . . . +a1tr+1 with the constant coefficients al−r=1, ai≥0 for i=1, 2,..., l−r−1. The bounds of λn+1 are expressed in terms of the preceding eigenvalues. This generalizes the inequalities obtained by Payne, Polya, Weinberger, Protter, Hile, and Yeh.