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

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].