Optimal unique continuation for periodic elliptic equations on large scales

Abstract

We prove a quantitative, large-scale doubling inequality and large-scale three-ellipsoid inequality for solutions of uniformly elliptic equations with periodic coefficients. These estimates are optimal in terms of the minimal length scale on which they are valid, and are at least "almost" optimal in the prefactor constants-up to, at most, an iterated logarithm of the initial doubling ratio.

Optimal Approximation of Unique Continuation

“…Definition 1 Assume that u ∈ H k+1 ( ), k ∈ N, solves the unique continuation problem (2). Let α ∈ (0, 1) be the largest value for which the conditional stability estimate (1) holds.…”

“…Proposition 1 Let (u h , z h ) denote the solution to(38) and let u ∈ H k+1 ( ) be the solution to(2), then there holds|||(u − u h , z h )||| S h k u H k+1 ( ) + δq L 2 (ω) .Proof First we decompose the error u − u h = u − h u + h u − u h =:e h in the continuous and discrete parts.…”

Higher-order boundary layers and regularity for Stokes systems over rough boundaries

“…The key to deriving the higher-order analog of (S) is to introduce the so-called boundary layers, which play similar roles as the correctors in elliptic homogenization with oscillating coefficients; see e.g., [9,4,31,22,11,7,6]. The first-order boundary layers are classical, given by the bounded solutions of the following cell problem: for i ∈ {1, 2, • • • , d},…”