Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

Asymptotic expansions and unique continuation at Dirichlet-Neumann boundary junctions for planar elliptic equations

1 African Institute for Mathematical Sciences (A.I.M.S.) of Senegal, KM 2, Route de Joal, AIMS-Senegal, B.P. 1418. Mbour, Senegal
2 Università di Milano-Bicocca, Dipartimento di Scienza dei Materiali, Via Cozzi 55, 20125 Milano,Italy
3 Università degli Studi del Piemonte Orientale, Dipartimento di Scienze e Innovazione Tecnologica, Viale Teresa Michel 11, 15121 Alessandria, Italy
4 Université Cheikh Anta Diop, BP 16 889 Dakar-Fann, Senegal

We consider elliptic equations in planar domains with mixed boundary conditions of Dirichlet-Neumann type. Sharp asymptotic expansions of the solutions and unique continuationproperties from the Dirichlet-Neumann junction are proved.
  Article Metrics

Keywords mixed boundary conditions; unique continuation; asymptotic expansion; monotonicity formula

Citation: Mouhamed Moustapha Fall, Veronica Felli, Alberto Ferrero, Alassane Niang. Asymptotic expansions and unique continuation at Dirichlet-Neumann boundary junctions for planar elliptic equations. Mathematics in Engineering, 2019, 1(1): 84-117. doi: 10.3934/Mine.2018.1.84


  • 1. Adolfsson V, Escauriaza L (1997) C1,α domains and unique continuation at the boundary. Comm Pure Appl Math 50: 935–969.    
  • 2. Adolfsson V, Escauriaza L, Kenig C (1995) Convex domains and unique continuation at the boundary. Rev Mat Iberoam 11: 513–525.
  • 3. Brändle C, Colorado E, de Pablo A, et al. (2013) A concave-convex elliptic problem involving the fractional Laplacian. Proceedings of the Royal Society of Edinburgh Section A: Mathematics 143: 39–71.    
  • 4. Caffarelli L, Silvestre L (2007) An extension problem related to the fractional Laplacian. Commun Part Diff Eq 32: 1245–1260.    
  • 5. Carleman T (1939) Sur un problème d'unicité pur les systèmes d'équations aux dérivées partielles à deux variables indépendantes. Ark Mat Astr Fys 26: 9.
  • 6. 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.    
  • 7. Doktor P, Zensek A (2006) The density of infinitely differentiable functions in Sobolev spaces with mixed boundary conditions. Appl Math 51: 517–547.    
  • 8. Fall MM, Felli V (2014) Unique continuation property and local asymptotics of solutions to fractional elliptic equations. Commun Part Diff Eq 39: 354–397.    
  • 9. Felli V, Ferrero A (2013) Almgren-type monotonicity methods for the classification of behaviour at corners of solutions to semilinear elliptic equations. Proceedings of the Royal Society of Edinburgh Section A: Mathematics 143: 957–1019.    
  • 10. Felli V, Ferrero A (2014) On semilinear elliptic equations with borderline Hardy potentials. J Anal Math 123: 303–340.    
  • 11. Felli V, Ferrero A, Terracini S (2011) Asymptotic behavior of solutions to Schrödinger equations near an isolated singularity of the electromagnetic potential. J Eur Math Soc 13: 119–174.
  • 12. Felli V, Ferrero A, Terracini S (2012) On the behavior at collisions of solutions to Schrödinger equations with many-particle and cylindrical potentials. Discrete Contin Dyn-A 32: 3895–3956.    
  • 13. 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.    
  • 14. Garofalo N, Lin FH (1986) Monotonicity properties of variational integrals, Ap weights and unique continuation. Indiana U Math J 35: 245–268.    
  • 15. Hartman P (1964) Ordinary Differential Equations, Wiley, New York.
  • 16. Kassmann M, Madych WR (2007) Difference quotients and elliptic mixed boundary value problems of second order. Indiana U Math J 56: 1047–1082.    
  • 17. Krantz SG (2006) Geometric function theory. Explorations in complex analysis, Cornerstones, Birkhäuser Boston, Inc., Boston, MA.
  • 18. Kukavica I (1998) Quantitative uniqueness for second-order elliptic operators. Duke Math J 91: 225–240.    
  • 19. Kukavica I, Nyström K (1998) Unique continuation on the boundary for Dini domains. Proc Amer Math Soc 126: 441–446.    
  • 20. Lazzaroni G, Toader R (2011) Energy release rate and stress intensity factor in antiplane elasticity. J Math Pures Appl 95: 565–584.    
  • 21. Ros-Oton X, Serra J (2014) The Dirichlet problem for the fractional Laplacian: regularity up to the boundary. J Math Pures Appl 101: 275–302.    
  • 22. Ros-Oton X, Serra J (2016) Regularity theory for general stable operators. J Differ Equations 260: 8675–8715.    
  • 23. Ros-Oton X, Serra J (2016) Boundary regularity for fully nonlinear integro-differential equations. Duke Math J 165: 2079–2154.    
  • 24. Serra J (2015) Regularity for fully nonlinear nonlocal parabolic equations with rough kernels. Calc Var Partial Dif 54: 615–629.    
  • 25. Savaré G (1997) Regularity and perturbation results for mixed second order elliptic problems. Commun Part Diff Eq 22: 869–899.    
  • 26. Tao X, Zhang S (2005) Boundary unique continuation theorems under zero Neumann boundary conditions. B Aust Math Soc 72: 67–85.    
  • 27. Tao X, Zhang S (2008) Weighted doubling properties and unique continuation theorems for the degenerate Schrödinger equations with singular potentials. J Math Anal Appl 339: 70–84.    
  • 28. Wolff TH (1992) A property of measures in R$^N$ and an application to unique continuation. Geom Funct Anal.


This article has been cited by

  • 1. Veronica Felli, Alberto Ferrero, Unique continuation and classification of blow-up profiles for elliptic systems with Neumann boundary coupling and applications to higher order fractional equations, Nonlinear Analysis, 2020, 196, 111826, 10.1016/j.na.2020.111826
  • 2. L. Abatangelo, V. Felli, C. Léna, Eigenvalue variation under moving mixed Dirichlet–Neumann boundary conditions and applications, ESAIM: Control, Optimisation and Calculus of Variations, 2020, 26, 39, 10.1051/cocv/2019022

Reader Comments

your name: *   your email: *  

© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved