Unique continuation from the edge of a crack

• Main Authors: Alessandra De Luca, Veronica Felli
• Format: Article
• Language: English
• Published: AIMS Press 2021-03-01
• Series: Mathematics in Engineering
• Subjects: crack singularities; monotonicity formula; unique continuation; blow-up analysis
• Online Access: http://www.aimspress.com/article/doi/10.3934/mine.2021023?viewType=HTML

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.