Large‐Scale Analyticity and Unique Continuation for Periodic Elliptic Equations

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 $O(\exp(\delta|x|))$ for small~$\delta>0$. 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.