Large‐Scale Analyticity and Unique Continuation for Periodic Elliptic Equations

Abstract

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 largescale 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.jxj// for small > 0. The large-scale analyticity also implies quantitative unique continuation results, namely a threeball theorem with an optimal error term as well as a proof of the nonexistence of L 2 eigenfunctions at the bottom of the spectrum.

Optimal unique continuation for periodic elliptic equations on large scales

“…Therefore, one may expect to obtain analogues of (1.1) and (1.3) for solutions periodic operators, at least on large enough length scales. Such results have indeed been proved recently by various authors [12,8,2,9,16]. However, the estimates in these works have been either qualitative (the dependence of various constants in the estimates is implicit) or else quantitative, but sub-optimal, in terms of the range of length scales on which they are valid, or else in terms of estimates for the constant C(M) in (1.1) or the parameters (α, C) in (1.2).…”

“…In that paper, we also proved a large-scale three-ball theorem as a corollary of large scale analyticity (see [2,Theorem 1.4]) down to the optimal scale C log M. However, in this estimate has sub-optimal exponent (like α < 1 /2 in (1.2)) and for this reason the estimate is not particularly useful (and cannot for instance be iterated to yield a doubling inequality without catastrophically blowing up the doubling ratio). Recently, Kenig, Zhu and Zhuge [9] proved, by argument which was also based on [2], a doubling estimate down to scale r = M δ for arbitrary δ > 0; this is, however, still far from the optimal scale of C log M, and their estimate for the doubling ratio is also much larger than the right side of (1.10). As explained below, the scales of order log M correspond to exponential growth, which is the critical regime for spectral theory problems.…”

“…The main ingredient in the proof of Theorem 1.1 is the large-scale analyticity estimate proved in our previous paper [2]. This result is the "best possible" quantitative estimate in periodic homogenization, and states essentially that corrector expansions of arbitrary high order can approximate solutions of a periodic elliptic equation up to an error which is exponentially small in the aspect ratio.…”

“…This is routine and has essentially already been done 1 in [13], so we omit the details. Since Corollary 1.3 was proved already in [2] by a simple argument which is similar to the one implied here, it has also been left to the reader. The set of natural numbers is N and N 0 = N ∪ {0}.…”

“…The first quantitative unique continuation results in the context of periodic homogenization was proved by Lin and Shen [12], who obtained a doubling inequality like Corollary 1.2 but with implicit dependence of the right side of (1.11) on the initial doubling ratio M. Kenig and Zhu [8] subsequently proved a version of the three-ball inequality (1.9) on scales r ≥ M C with sub-optimal exponents. In our previous paper [2], we improved this result by using large-scale analyticity to obtain a a three-ball inequality is valid down to scale C log M with almost optimal exponents. Later, Kenig, Zhu and Zhuge [9, Theorem 1.1] proved, by argument also based on large-scale analyticity, a similar estimate to (1.11) with a right side which is doubly exponential in M, namely exp(exp(M δ )) for small δ > 0.…”