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Oscillation theorems for elliptic differential equations Headley, Velmer Bentley

Abstract

Criteria will be obtained for a linear self-adjoint elliptic partial differential equation to be oscillatory or nonoscillatory in unbounded domains R of n-dimensional Euclidean space Eⁿ. The criteria are of two main types: (i) those involving integrals of suitable majorants of the coefficients, and (ii) those involving limits of these majorants as the argument tends to infinity. Our theorems constitute generalizations to partial differential equations of well-known criteria of Hille, Leighton, Potter, Moore, and Wintner for ordinary differential equations. In general, our method provides the means for extending in this manner any oscillation criterion for self-adjoint ordinary differential equations. Our results imply Glazman's theorems in the special case of the Schrodinger equation in Eⁿ. In the derivation of the oscillation criteria it is assumed that R is either quasiconical (i.e. contains an infinite cone) or limit-cylindrical (i.e. contains an infinite cylinder). In the derivation of the nonoscillation criteria no special assumptions regarding the shape of the domain are needed. Examples illustrating the theory are given. In particular, it is shown that the limit criteria obtained in the second order case are the best possible of their kind.

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