Asymptotic analysis for elliptic systems in a narrow region

TL;DR

This paper studies the asymptotic behavior of solutions to elliptic systems in narrow regions, providing precise gradient estimates and correction terms to better understand singularities, especially for Lamé systems.

Contribution

It introduces a correction term that refines previous gradient estimates, offering a sharp asymptotic characterization of singularities in elliptic systems within narrow regions.

Findings

Derived a precise asymptotic formula for gradients

Captured all singular terms in the asymptotic analysis

Provided sharp characterization of gradient singularities

Abstract

This paper is concerned with investigating the asymptotic behavior of the gradients of solutions to a class of elliptic systems with general boundary data, especially covering the Lam\'{e} systems, in a narrow region. The novelty of this paper lies in finding the correction term to improve the previous gradient estimates to be a precise asymptotic formula. Through this justification, we capture all singular terms and thus give a fairly sharp characterization in terms of the singularities of the gradients.