### 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.

 [1] Adolfsson V (1992) L2-integrability of second-order derivatives for Poisson's equation in nonsmooth domains. Math Scand 70: 146-160. [2] Adolfsson V, Escauriaza L (1997) C1, α domains and unique continuation at the boundary. Commun Pure Appl Math 50: 935-969. [3] Adolfsson V, Escauriaza L, Kenig C (1995) Convex domains and unique continuation at the boundary. Rev Mat Iberoam 11: 513-525. [4] Almgre Jr FJ (1983) Q valued functions minimizing Dirichlet's integral and the regularity of area minimizing rectifiable currents up to codimension two. B Am Math Soc 8: 327-328. [5] Bernard JME (2011) Density results in Sobolev spaces whose elements vanish on a part of the boundary. Chinese Ann Math B 32: 823-846. [6] Carleman T (1939) Sur un problème d'unicité pur les systèmes d' équations aux dérivées partielles à deux variables indéependantes. Ark Mat Astr Fys 26: 9. [7] Chkadua O, Duduchava R (2000) Asymptotics of functions represented by potentials. Russ J Math Phys 7: 15-47. [8] Costabel M, Dauge M, Duduchava R (2003) Asymptotics without logarithmic terms for crack problems. Commun Part Diff Eq 28: 869-926. [9] Dal Maso G, Orlando G, Toader R (2015) Laplace equation in a domain with a rectilinear crack: higher order derivatives of the energy with respect to the crack length. NoDEA Nonlinear Diff 22: 449-476. [10] Daners D (2003) Dirichlet problems on varying domains. J Differ Equations 188: 591-624. [11] Dipierro S, Felli V, Valdinoci E (2020) Unique continuation principles in cones under nonzero Neumann boundary conditions. Ann I H Poincaré Anal non linéaire 37: 785-815. [12] Duduchava R, Wendland WL (1995) The Wiener-Hopf method for systems of pseudodifferential equations with an application to crack problems. Integr Equat Oper Th 23: 294-335. [13] Fabes EB, Garofalo N, Lin FH (1990) A partial answer to a conjecture of B. Simon concerning unique continuation. J Funct Anal 88: 194-210. [14] Fall MM, Felli V, Ferrero A, et al. (2019) Asymptotic expansions and unique continuation at Dirichlet-Neumann boundary junctions for planar elliptic equations. Mathematics in Engineering 1: 84-117. [15] Felli V, Ferrero A (2013) Almgren-type monotonicity methods for the classification of behaviour at corners of solutions to semilinear elliptic equations. P Roy Soc Edinb A 143: 957-1019. [16] Felli V, Ferrero A (2014) On semilinear elliptic equations with borderline Hardy potentials. J Anal Math 123: 303-340. [17] Felli V, Ferrero A, Terracini S (2011) Asymptotic behavior of solutions to Schr?dinger equations near an isolated singularity of the electromagnetic potential. J Eur Math Soc 13: 119-174. [18] Felli V, Ferrero A, Terracini S (2012) A note on local asymptotics of solutions to singular elliptic equations via monotonicity methods. Milan J Math 80: 203-226. [19] Felli V, Ferrero A, Terracini S (2012) On the behavior at collisions of solutions to Schr?dinger equations with many-particle and cylindrical potentials. Discrete Contin Dyn Syst 32: 3895-3956. [20] Garofalo N, Lin FH (1986) Monotonicity properties of variational integrals, Ap weights and unique continuation. Indiana U Math J 35: 245-268. [21] Kassmann M, Madych WR (2007) Difference quotients and elliptic mixed boundary value problems of second order. Indiana U Math J 56: 1047-1082. [22] Khludnev A, Leontiev A, Herskovits J (2003) Nonsmooth domain optimization for elliptic equations with unilateral conditions. J Math Pure Appl 82: 197-212. [23] Kukavica I (1998) Quantitative uniqueness for second-order elliptic operators. Duke Math J 91: 225-240. [24] Kukavica I, Nyström K (1998) Unique continuation on the boundary for Dini domains. P Am Math Soc 126: 441-446. [25] Lazzaroni G, Toader R (2011) Energy release rate and stress intensity factor in antiplane elasticity. J Math Pure Appl 95: 565-584. [26] Mosco U (1969) Convergence of convex sets and of solutions of variational inequalities. Adv Math 3: 510-585. [27] Savaré G (1997) Regularity and perturbation results for mixed second order elliptic problems. Commun Part Diff Eq 22: 869-899. [28] Tao X, Zhang S (2005) Boundary unique continuation theorems under zero Neumann boundary conditions. B Aust Math Soc 72: 67-85. [29] Tao X, Zhang S (2008) Weighted doubling properties and unique continuation theorems for the degenerate Schr?dinger equations with singular potentials. J Math Anal Appl 339: 70-84. [30] Wang ZQ, Zhu M (2003) Hardy inequalities with boundary terms. Electron J Differ Eq 2003: 1-8. [31] Wolff TH (1992) A property of measures in $\mathbb{R}.N$ and an application to unique continuation. Geom Funct Anal 2: 225-284.
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