Comparison theorems for elliptic differential equations

Abstract

In recent years the classical Sturm-Picone comparison theorem for a pair of second order ordinary differential equations has been generalized in two directions. First, Hartman and Wintner [3] have extended the theorem to self-adjoint elliptic equations in n dimensions. On the other hand, Leighton [5] has shown in the case n = 1 that the usual pointwise inequalities for the coefficients can be replaced by a more general inequality. Our main purpose in this note is to extend Leighton's result to self-adjoint elliptic equations. The HartmanWintner result is a corollary of our main theorem. We also give an example to show that our result is actually stronger. An interesting feature is the simplicity of our proof as compared to that of Hartman and Wintner or the recent proof of Kreith [4]. Our result, like Leighton's, is an easy consequence of a theorem from the calculus of variations. A simple proof of the latter theorem in the form required is outlined below; it depends only on Green's formula and an elementary identity. Let R be a bounded, open set in n-dimensional Euclidean space En, with boundary B having a piecewise continuous unit normal. A typical point of En will be denoted by x = (xl, x2, * * *, x8). Partial differentiation with respect to xi will be denoted by Di (i = 1, 2, * * , n). The linear, elliptic self -adjoint partial differential operator L defined by