Unique continuation from a crack's tip under Neumann boundary conditions

TL;DR

This paper analyzes the behavior of solutions to elliptic equations near crack tips with Neumann boundary conditions, establishing asymptotic classifications and a strong unique continuation principle.

Contribution

It introduces a novel classification of asymptotic behaviors at crack tips and proves a strong unique continuation principle for elliptic equations with Neumann conditions.

Findings

Classified all possible asymptotic homogeneities at crack tips

Established a strong unique continuation principle

Used blow-up analysis and monotonicity methods

Abstract

We derive local asymptotics of solutions to second order elliptic equations at the edge of a $(N-1)$-dimensional crack, with homogeneous Neumann boundary conditions prescribed on both sides of the crack. A combination of blow-up analysis and monotonicity arguments provides a classification of all possible asymptotic homogeneities of solutions at the crack's tip, together with a a strong unique continuation principle.