1. Unique continuation from the edge of a crack
- Author
-
Veronica Felli, Alessandra De Luca, De Luca, A, and Felli, V
- Subjects
blow-up analysis ,Mathematics::Analysis of PDEs ,crack singularities ,Monotonic function ,Edge (geometry) ,Type (model theory) ,Domain (mathematical analysis) ,Continuation ,Mathematics - Analysis of PDEs ,Crack singularitie ,FOS: Mathematics ,Point (geometry) ,Limit (mathematics) ,MAT/05 - ANALISI MATEMATICA ,Mathematical Physics ,Mathematics ,Sequence ,Applied Mathematics ,lcsh:T57-57.97 ,Mathematical analysis ,35J15, 35C20, 74A45 ,unique continuation ,Blow-up analysi ,monotonicity formula ,lcsh:Applied mathematics. Quantitative methods ,Analysis ,Analysis of PDEs (math.AP) - 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., 32 pages, 2 figures
- Published
- 2021