### Mathematics in Engineering

2021, Issue 3: 1-40. doi: 10.3934/mine.2021023
Research article Special Issues

# Unique continuation from the edge of a crack

• Received: 23 April 2020 Accepted: 29 June 2020 Published: 16 July 2020
• 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.

Citation: Alessandra De Luca, Veronica Felli. Unique continuation from the edge of a crack[J]. Mathematics in Engineering, 2021, 3(3): 1-40. doi: 10.3934/mine.2021023

### Related Papers:

• 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.

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