Published Paper

Inserted: 9 sep 2024
Last Updated: 9 sep 2024

Journal: Mathematics in Engineering
Year: 2020
Doi: https://doi.org/10.3934/mine.2021023

Abstract:

In this work we develop an Almgren type monotonicity formula for a class of elliptic equations in a domain with a crack, in the presence of potentials satisfying either a negligibility condition with respect to the inverse-square weight or some suitable integrability properties. The study of the Almgren frequency function around a point on the edge of the crack, where the domain is highly non-smooth, requires the use of an approximation argument, based on the construction of a sequence of regular sets which approximate the cracked domain. Once a finite limit of the Almgren frequency is shown to exist, a blow-up analysis for scaled solutions allows us to prove asymptotic expansions and strong unique continuation from the edge of the crack.