A remark on a comparison theorem of Swanson

Abstract

defined in a domain P with piecewise continuous unit normal on the boundary B. Given that F is a strict Sturmian majorant of L* and that there exists a nontrivial solution of (1) satisfying m = 0 on B, Swanson shows that every solution of (2) has a zero in This result is not in the sense of [2] where it is shown that under similar hypotheses in the selfadjoint case, every solution of (2) must vanish in the interior of The purpose of this note is to point out that if B is of bounded curvature, then one can use the method of [l] to arrive at this stronger conclusion even in the nonselfadjoint case. Specifically, if it can be shown that every solution of (2) which is not zero in R and vanishes at a point x0£F> must satisfy (dv/dv) (x0) 5^0, then it is clear from the proof that the Lemma of [l] can be altered to read: Suppose g satisfies g det(a,j)> — £?-i bist. If there exists m£i> not identically zero such that /[w];£0, then every solution v of Lv = 0 vanishes at some point of R. A strong version of Swanson's comparison theorem follows readily from this change, and in the case of ordinary differential equations (i.e. n = 1) this fact is observed in [l ]. If c^O near B and B is of bounded curvature, then it follows from the Hopf maximum principle [3] that (dv/dv) (x0) ?* 0 whenever i>(xo)=0, xo£P. However, even if c is merely bounded, the same conclusion can be derived. To see this we assume v <0 in R and v(x0) =0 for some x0£F. Without loss of generality we may assume that B is tangent to the plane Xi = b and that the exterior normal to B at x0 is in the positive indirection. It is known (see [4, p. 73]) that for (b—a) sufficiently small there exist positive constants a, s such that