Über das Spektrum des Laplace-Operators in einigen unbeschränkten Gebieten

Abstract

In this paper we study the eigenvalues of the Laplace operator Δ u on some classes of unbounded regions G. The starting-point of our study is the paper [5] of Vogelsang. Among other things he has proved that the Laplace operator with zero boundary condition has no negative eigenvalues in unbounded regions the boundary of which satisfies ν(x) · x ≦0, where ν(x) is the exterior normal. We prove in this paper similar results of the spectrum, when the condition above is suitably disturbed. Because our main interest lies in replacing the geometric condition ν(x) · x ≦0 with another condition, we have studied neither equations with higher orders nor equations with variable coefficients.