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	NCTS Seminar on PDE and Analysis

14:20 - 15:20, January 5, 2017 (Thursday)
R440, Astronomy-Mathematics Building, NTU
(台灣大學天文數學館 440室)
Recent Progress on Landis’ Conjecture
Blair Davey (The City College of New York)

Abstract

In the late 1960s, E.M. Landis made the following conjecture:

If $u$ and $V$ are bounded functions, and $u$ is a solution to $\Delta u = V u$ in $\mathbb{R}^n$ that decays like $|u(x)| \le c \exp(- C |x|^{1+})$ , then $u$ must be identically zero. In 1992, V. Z. Meshkov disproved this conjecture by constructing bounded functions $u, V: \mathbb{R}^2 \to \mathbb{C}$ that solve$ \Delta u = V u $in $\mathbb{R}^2$ and satisfy $|u(x)| \le c \exp(- C |x|^{4/3})$ .

The result of Meshkov was accompanied by qualitative unique continuation estimates for solutions in $\mathbb{R}^n$ . In 2005, J. Bourgain and C. Kenig quantified Meshkov's unique continuation estimates.These results, and the generalizations that followed, have led to a fairly complete understanding of the complex-valued setting.However, there are reasons to believe that Landis' conjecture may be true in the real-valued setting.We will discuss recent progress towards resolving the real-valued version of Landis' conjecture in the plane.