arXiv 1807.003412018-07math.AP

具有锐衰变率的Landis猜想

The Landis conjecture with sharp rate of decay

Luca Rossi

摘要：

所谓的Landis猜想指出，如果方程$$Δu+V(x)u=0$$在外部区域中的解衰减得比$e^{-κ|x|}$快，对于某些$κ>\sqrt{\sup |V|}$，则它必须与$0$相同。这个性质可以被看作是一个唯一的无穷远延拓（UCI）的解决方案，满足适当的指数衰减。兰迪斯猜想在复值函数的情况下被梅什科夫证伪，但在实际情况下仍然是开放的。在2000年代，一些论文讨论了实系数线性椭圆算子的UCI问题。已经得到的结果需要某种符号条件，或者是关于方程的解，或者是关于方程的零级系数。Landis猜想的一般形式至今仍是公开的。在本文中，我们首先考虑一个一般的（实）椭圆算子的维数$1$。我们推导出UCI性质的衰减率$κ$是尖锐的系数时，运营商是常数。特别地，我们在$1$维上证明了Landis猜想，并且我们实际上可以达到门限值$κ=\sqrt{\sup |V|}$ .接下来，我们推导出任意维径向算子的UCI性质--然后是Landis猜想。最后，我们用另一种方法证明了一般椭圆型方程正上解的同样结果。

英文摘要：

The so called Landis conjecture states that if a solution of the equation $$Δu+V(x)u=0$$ in an exterior domain decays faster than $e^{-κ|x|}$, for some $κ>\sqrt{\sup |V|}$, then it must be identically equal to $0$. This property can be viewed as a unique continuation at infinity (UCI) for solutions satisfying a suitable exponential decay. The Landis conjecture was disproved by Meshkov in the case of complex-valued functions, but it remained open in the real case. In the 2000s, several papers have addressed the issue of the UCI for linear elliptic operators with real coefficients. The results that have been obtained require some kind of sign condition, either on the solution or on the zero order coefficient of the equation. The Landis conjecture is still open nowadays in its general form. In the present paper, we start with considering a general (real) elliptic operator in dimension $1$. We derive the UCI property with a rate of decay $κ$ which is sharp when the coefficients of the operator are constant. In particular, we prove the Landis conjecture in dimension $1$, and we can actually reach the threshold value $κ=\sqrt{\sup |V|}$. Next, we derive the UCI property -- and then the Landis conjecture -- for radial operators in arbitrary dimension. Finally, with a different approach, we prove the same result for positive supersolutions of general elliptic equations.

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