Unique continuation from the edge of a crack

Alessandra De Luca; Veronica Felli; Alessandra De Luca; Veronica Felli

doi:10.3934/mine.2021023

Mathematics in Engineering

2021, Volume 3, Issue 3: 1-40. doi: 10.3934/mine.2021023

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Unique continuation from the edge of a crack

Alessandra De Luca ^,,
Veronica Felli ^,

Dipartimento di Matematica e Applicazioni, Università di Milano - Bicocca, Via Cozzi 55, 20125 Milano, Italy

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.
- crack singularities,
- monotonicity formula,
- unique continuation,
- blow-up analysis
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

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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.

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