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.


    加载中


    [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.
  • Reader Comments
  • © 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(332) PDF downloads(138) Cited by(0)

Article outline

Figures and Tables

Figures(2)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog