论文标题
周期性椭圆方程的大规模分析性和独特的延续
Large-scale analyticity and unique continuation for periodic elliptic equations
论文作者
论文摘要
我们证明，具有周期性系数的椭圆运算符的解在诸如分析函数之类的大尺度上表现出来，从而通过多项式进行了定期校正的近似。同等地，大规模$ c^{k，1} $估计比例为$ k $，就像谐波功能的经典估算一样。结果，我们表征了整个周期性，均匀椭圆方程的解决方案，这些方程式表现出生长，例如$ o（\ exp（δ| x |））$，对于小〜$δ> 0 $。大规模的分析性还意味着定量独特的延续结果，即具有最佳误差项的三球定理以及频谱底部$ l^2 $ eigenfunctions不存在的证明。
We prove that a solution of an elliptic operator with periodic coefficients behaves on large scales like an analytic function, in the sense of approximation by polynomials with periodic corrections. Equivalently, the constants in the large-scale $C^{k,1}$ estimate scale exponentially in $k$, just as for the classical estimate for harmonic functions. As a consequence, we characterize entire solutions of periodic, uniformly elliptic equations which exhibit growth like $O(\exp(δ|x|))$ for small~$δ>0$. The large-scale analyticity also implies quantitative unique continuation results, namely a three-ball theorem with an optimal error term as well as a proof of the nonexistence of $L^2$ eigenfunctions at the bottom of the spectrum.
