A note on geometric assumptions for unique continuation from the edge of a crack

TL;DR

This paper improves unique continuation results for elliptic problems near crack edges by removing star-shapedness assumptions through boundary straightening and monotonicity formulas, enhancing understanding of local behavior.

Contribution

It introduces a method to relax geometric assumptions in unique continuation problems using boundary straightening and matrix handling techniques.

Findings

Relaxed geometric conditions for unique continuation.

Applied boundary straightening to elliptic crack problems.

Derived new monotonicity formulas accommodating matrix terms.

Abstract

The present paper aims at representing an improvement of the result in [2], where a strong unique continuation property and a description of the local behaviour around the edge of a crack for solutions to an elliptic problem are established, by relaxing the star-shapedness condition on the complement of the crack. More specifically, this assumption will be dropped off by applying a suitable diffeomorphism which straightens the boundary of the crack, before performing the approximation procedure developed in [2] in order to derive a suitable monotonicity formula. This will yield the appearence of a matrix in the equation, which shall be handled appropriately: for this we will take a hint from [4].