By Bruno Bianchini
The purpose of this paper is to examine a few of the relationships among oscillation idea for linear usual differential equations at the actual line (shortly, ODE) and the geometry of whole Riemannian manifolds. With this motivation the authors end up a few new leads to either instructions, starting from oscillation and nonoscillation stipulations for ODE's that increase on classical standards, to estimates within the spectral thought of a few geometric differential operator on Riemannian manifolds with comparable topological and geometric functions. to maintain their research primarily self-contained, the authors additionally acquire a few, roughly recognized, fabric which regularly seems to be within the literature in numerous types and for which they offer, in a few situations, new proofs in line with their particular viewpoint
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Additional resources for On some aspects of oscillation theory and geometry
The characterization of the bottom of the essential spectrum is due to A. Persson in the previous paper [Per60]. We shall remark that the proofs in [BCS97] and [PRS08] require not only that L is bounded from below, but also that (L, Cc∞ (M )) is essentially self-adjoint. This condition, automatic if M is complete and L = −Δ − q(x), turns out to be quite 36 2. 5). 81) in not automatically satisﬁed. However, reﬁning the proof in [PRS08], in the recent [Mar] the author has observed that both the completeness assumption on M and the essential self-adjointness of (L, Cc∞ (M )) are, in fact, unnecessary.
34). Moreover, if s1 < +∞, B cannot be deﬁned past s1 . Indeed, let X be a unit parallel vector ﬁeld such that JX(s1 ) = 0. Then, since JX ≡ 0, (JX) (s1 ) = 0. 39) BJX, JX J X, JX 1 d log |JX|2 → −∞ = = |JX|2 |JX|2 2 ds as s → s− 1. This means that the function hess r ◦ γ can be extended past the cut-point of o along γ, if the cut-point is non-focal, and the maximal extension is deﬁned on (0, s1 ), where γ(s1 ) is the ﬁrst focal point of o along γ. At γ(s1 ), however, hess r ◦ γ presents a singularity, and more precisely it is unbounded from below as s → s1 .
Applying the weak deﬁnition of Lu = λL 1 (Ω)u to the test functions u+ , and using the min-max deﬁnition of λL 1 (Ω) we get L 0 = QΩ − λL 1 (Ω)( , ) (u, u+ ) ≡ QΩ − λ1 (Ω)( , ) (u+ , u+ ) ≥ 0, thus u+ is a minimum of the Rayleigh quotient. Analogously, we can prove that also u− is a minimum. Hence, by Courant minimum principle u+ and u− are both eigenfunctions, each vanishing on some nonempty open subset of Ω. This contradicts the unique continuation property ([Aro57] and [PRS08], Appendix A). Up to changing the sign, this shows that u ≥ 0 on Ω.
On some aspects of oscillation theory and geometry by Bruno Bianchini