On Landis’ conjecture in the plane
نویسندگان
چکیده
In this paper we prove a quantitative form of Landis’ conjecture in the plane. Precisely, let W (z) be a measurable real vector-valued function and V (z) ≥ 0 be a real measurable scalar function, satisfying ‖W‖L∞(R2) ≤ 1 and ‖V ‖L∞(R2) ≤ 1. Let u be a real solution of ∆u − ∇(Wu) − V u = 0 in R2. Assume that u(0) = 1 and |u(z)| ≤ exp(C0|z|). Then u satisfies inf |z0|=R sup |z−z0|<1 |u(z)| ≥ exp(−CR logR), where C depends on C0. In addition to the case of the whole plane, we also establish a quantitative form of Landis’ conjecture defined in an exterior domain.
منابع مشابه
On the oriented perfect path double cover conjecture
An oriented perfect path double cover (OPPDC) of a graph $G$ is a collection of directed paths in the symmetric orientation $G_s$ of $G$ such that each arc of $G_s$ lies in exactly one of the paths and each vertex of $G$ appears just once as a beginning and just once as an end of a path. Maxov{'a} and Ne{v{s}}et{v{r}}il (Discrete Math. 276 (2004) 287-294) conjectured that ...
متن کاملA note on Fouquet-Vanherpe’s question and Fulkerson conjecture
The excessive index of a bridgeless cubic graph $G$ is the least integer $k$, such that $G$ can be covered by $k$ perfect matchings. An equivalent form of Fulkerson conjecture (due to Berge) is that every bridgeless cubic graph has excessive index at most five. Clearly, Petersen graph is a cyclically 4-edge-connected snark with excessive index at least 5, so Fouquet and Vanherpe as...
متن کامل$L^p$-Conjecture on Hypergroups
In this paper, we study $L^p$-conjecture on locally compact hypergroups and by some technical proofs we give some sufficient and necessary conditions for a weighted Lebesgue space $L^p(K,w)$ to be a convolution Banach algebra, where $1<p<infty$, $K$ is a locally compact hypergroup and $w$ is a weight function on $K$. Among the other things, we also show that if $K$ is a locally compact hyper...
متن کامل