An identity for elliptic equations with applications
Charles A. Swanson · Transactions of the American Mathematical Society · 1968
SWANSONO 1. Introduction.An elementary identity involving a linear elliptic partial differential operator L and its associated hermitian form will be used to obtain new comparison theorems, oscillation theorems, and lower bounds for eigenvalues.Comparison theorems will be obtained for both subsolutions and complex-valued solutions in unbounded domains of Euclidean space, generalizing earlier results of Hartman and Wintner [4], Protter [8], and the author [11], [12].Oscillation theorems of Kreith's type [6] will be extended to (i) unbounded domains; (ii) nonself-adjoint operators ; and (iii) subsolutions.Lower bounds for the eigenvalues of L arise naturally from the basic identity in the case of bounded domains, and are extended to unbounded domains when the coefficients of L satisfy suitable conditions.The form of the lower bounds is the same as that obtained by Protter and Weinberger [9], [10] for bounded domains.