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, and the minimal scale grows at most linearly in k. As a consequence, we characterize entire solutions of periodic, uniformly elliptic equations that 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 L² eigenfunctions at the bottom of the spectrum. © 2020 Wiley Periodicals LLC.

中文翻译：

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周期椭圆方程的大规模解析性和唯一连续性

我们证明了具有周期性系数的椭圆算子的解在大尺度上的行为类似于具有周期性校正的多项式逼近意义上的解析函数。等效地，大规模C ^{k , 1中的常数估计在}k中呈指数增长，就像调和函数的经典估计一样，最小规模在k中至多呈线性增长。因此，我们描述了周期性均匀椭圆方程的整个解，这些方程表现出像O (exp( δ |  x | ))对于小δ  > 0的增长. 大规模解析性还意味着定量的唯一延拓结果，即具有最优误差项的三球定理以及谱底L ^{2本征函数不存在的证明。}© 2020 威利期刊有限责任公司。