Nodal Sets and Doubling Conditions in Elliptic Homogenization

Abstract

This paper is concerned with uniform measure estimates for nodal sets of solutions in elliptic homogenization. We consider a family of second-order elliptic operators {L ε } in divergence form with rapidly oscillating and periodic coefficients. We show that the (d − 1)-dimensional Hausdorff measures of the nodal sets of solutions to L ε (u ε ) = 0 in a ball in R d are bounded uniformly in ε > 0. The proof relies on a uniform doubling condition and approximation of u ε by solutions of the homogenized equation.

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).…”

“…Comparison to previous works. 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.…”

A Carleman-Type Inequality in Elliptic Periodic Homogenization

“…The Corollary above only implies the three-ball inequality at a macroscopic scale, in the following theorem, we could obtain the three-ball inequality at every scale by using Corollary 1.4 and the uniform doubling conditions proved in [18] in elliptic homogenization.…”

“…Recently, authors in [2,15,16,18] care about the propagation of smallness in homogenization theory, such as the approximate three-ball inequality in [2,15] in elliptic periodic homogenization and the approximate two-sphere one-cylinder inequality in [24] in parabolic case, and the nodal sets and doubling conditions in [16,18] in elliptic homogenization, which are all related to the Carleman inequality in classical elliptic and parabolic theory and encourage us to deduce a Carleman-type inequality in elliptic periodic homogenization and left for further for the parabolic case.…”

A New Estimate for the Homogenization Method for Second-Order Elliptic Problem with Rapidly Oscillating Periodic Coefficients

“…There are numerous literatures discussing the homogenization method (see [1,2,[4][5][6][7][8][9]). There also are many works (see [3,[10][11][12][13][14][15][16]) discussing the numerical methods of the multiscale homogenization problem.…”