Asymptotic behaviour of the solutions of a non-linear transmission problem for the Laplace operator in a domain with a small hole. A functional analytic approach

References

• Mityushev , VV and Rogosin , SV . 2000 . “ Constructive Methods for Linear and Nonlinear Boundary Value Problems for Analytic Functions. Theory and Applications ” . In Monographs and Surveys in Pure and Applied Mathematics , London : Chapman &Hall/CRC .
   Google Scholar
• Kozlov , VA , Maz'ya , VG and Movchan , AB . 1999 . Asymptotic Analysis of Fields in Multistructures , New York : The Clarendon Press, Oxford University Press .
   Google Scholar
• Maz'ya , VG , Nazarov , SA and Plamenewskii , BA . 2000 . Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains, I, II, (translation of the original in German published by Akademie Verlag 1991) . Oper. Theory Adv. Appl. , 111, 112, Birkhäuser Verlag, Basel
   Google Scholar
• Movchan , AB . 1999 . “ Contributions of V.G. Maz'ya to Analysis of Singularly Perturbed Boundary Value Problems ” . In The Maz'ya Anniversary Collection, 1 (Rostock, 1998), Oper. Theory Adv. Appl , Vol. 109 , 201 – 212 . Basel : Birkhäuser .
   Google Scholar
• Ozawa , S . 1983 . Electrostatic capacity and eigenvalues of the Laplacian . J. Fac. Sci. Univ. Tokyo Sect. IA Math. , 30 : 53 – 62 .
   Google Scholar
• Vogelius , MS and Volkov , D . 2000 . Asymptotic formulas for perturbations in the electromagnetic fields due to the presence of inhomogeneities of small diameter . Math. Model. Numer. Anal. , 34 : 723 – 748 .
   Web of Science ®Google Scholar
• Ward , MJ and Keller , JB . 1993 . Strong localized perturbations of eigenvalue problems . SIAM J. Appl. Math. , 53 : 770 – 798 .
   Web of Science ®Google Scholar
• Ball , JM . 1982 . Discontinuous equilibrium solutions and cavitation in non-linear elasticity . Philos. Trans. Roy. Soc. London Ser. A , 306 : 557 – 611 .
   Google Scholar
• Sivaloganathan , J , Spector , SJ and Tilakraj , V . 2006 . The convergence of regularized minimizers for cavitation problems in non-linear elasticity . SIAM J. Appl. Math. , 66 : 736 – 757 .
   Web of Science ®Google Scholar
• Dal Maso , G and Murat , F . 2004 . Asymptotic behaviour and correctors for linear Dirichlet problems with simultaneously varying operators and domains . Ann. Inst. H. Poincarè Anal. Non Linéaire , 21 : 445 – 486 .
   Web of Science ®Google Scholar
• Titcombe , MS and Ward , MJ . 1999 . Summing logarithmic expansions for elliptic equations in multiply-connected domains with small holes . Canad. Appl. Math. Quart. , 7 : 313 – 343 .
   Google Scholar
• Ward , MJ , Henshaw , W and Keller , JB . 1993 . Summing logarithmic expansions for singularly perturbed eigenvalue problems . SIAM J. Appl. Math. , 53 : 799 – 828 .
   Web of Science ®Google Scholar
• Ward , MJ and Keller , JB . 1991 . Nonlinear eigenvalue problems under strong localized perturbations with applications to chemical reactors . Stud. Appl. Math. , 85 : 1 – 28 .
   Web of Science ®Google Scholar
• Lanza de Cristoforis , M . 2004 . “ Asymptotic behaviour of the conformal representation of a Jordan domain with a small hole, and relative capacity ” . In Contemporary Mathematics, Vol. 364 , Edited by: Agranovsky , M , Karp , L , Shoikhet , D and Zalcman , L . 155 – 167 . Providence , RI : Complex Analysis and Dynamical Systems, American Mathematical Society . (Proceeding Conference Karmiel, June 19–22, 2001)
   Google Scholar
• Lanza de Cristoforis , M . 2002 . Asymptotic behaviour of the conformal representation of a Jordan domain with a small hole in Schauder spaces . Comput. Meth. Funct. Theory , 2 : 1 – 27 .
   Google Scholar
• Lanza de Cristoforis , M . 2008 . Asymptotic behaviour of the solutions of the Dirichlet problem for the Laplace operator in a domain with a small hole. A functional analytic approach . Analysis (Munich) , 28 : 63 – 93 .
   Google Scholar
• Lanza de Cristoforis , M . 2005 . A singular domain perturbation problem for the Poisson equation . Complex Var. Theory Appl. , 50 : 851 – 867 .
   Google Scholar
• Lanza de Cristoforis , M . July 27–31 2004 . A singular perturbation Dirichlet boundary value problem for harmonic functions on a domain with a small hole , July 27–31 , Tokyo : in Proceedings of the 12th International Conference on Finite or Infinite Dimensional Complex Analysis and Applications . H. Kazama, M. Morimoto, C. Yang, eds., Kyushu University Press, Fukuoka, 2005, pp. 205–212
   Google Scholar
• Lanza de Cristoforis , M . 2007 . Asymptotic behaviour of the solutions of a non-linear Robin problem for the Laplace operator in a domain with a small hole: A functional analytic approach . Complex Var. Elliptic Equ. , 52 : 945 – 977 .
   Google Scholar
• Lanza de Cristoforis , M . 2007 . Singular perturbation problems in potential theory and applications, in Complex Analysis and Potential Theory (Proc. Conf. Satellite to ICM 2006, Gebze Institute of Technolgy, Turkey, September 8–14, 2006) , Edited by: Azeroğlu , TA and Tamrazov , PM . 131 – 139 . Hackensack , NJ : World Scientific Publication .
   Google Scholar
• Prodi , G and Ambrosetti , A . 1973 . Analisi Non Lineare , Pisa : Editrice Tecnico Scientifica .
   Google Scholar
• Gilbarg , D and Trudinger , NS . 1983 . Elliptic Partial Differential Equations of Second Order , Berlin : Springer Verlag .
   Google Scholar
• Lanza de Cristoforis , M . 1991 . Properties and Pathologies of the composition and inversion operators in Schauder spaces . Rend. Accad. Naz. Sci. XL , 15 : 93 – 109 .
   Google Scholar
• Lanza de Cristoforis , M and Rossi , L . 2004 . Real analytic dependence of simple and double layer potentials upon perturbation of the support and of the density . J. Int. Equ. Appl. , 16 : 137 – 174 .
   Google Scholar
• Folland , GB . 1995 . Introduction to Partial Differential Equations, , 2nd , Princeton , N.J. : Princeton University Press .
   Google Scholar
• Miranda , C . 1970 . Partial Differential Equations of Elliptic Type, , 2nd revised , New York-Berlin : Springer-Verlag .
   Google Scholar
• Miranda , C . 1965 . Sulle proprietà di regolarità di certe trasformazioni integrali . Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez I , 7 : 303 – 336 .
   Google Scholar
• Deimling , K . 1985 . Nonlinear Functional Analysis , Berlin : Springer-Verlag .
   Google Scholar
• Schauder , J . 1931 . Potentialtheoretische Untersuchungen . Math. Z. , 33 : 602 – 640 .
   Web of Science ®Google Scholar
• Böhme , R and Tomi , F . 1973 . Zur Struktur der Lösungsmenge des Plateauproblems . Math. Z. , 133 : 1 – 29 .
   Web of Science ®Google Scholar
• Henry , D . 1982 . Topics in Non-linear Analysis , Vol. 192 , Brasilia : Trabalho de Matemàtica, Universidade de Brasilia .
   Google Scholar
• Valent , T . 1988 . Boundary Value Problems of Finite Elasticity. Local Theorems on Existence, Uniqueness and Analytic Dependence on Data , New York : Springer-Verlag .
   Google Scholar
• E. Hille and R.S. Phillips, Functional analysis and semigroups, rev. ed. Am. Math. Soc. Colloq. Publ. vol 31, American Mathematical Society, Providence, RI, 1957
   Google Scholar
• Lanza de Cristoforis , M . 2007 . Perturbation problems in potential theory. A functional analytic approach . J. Appl. Funct. Anal , 2 : 197 – 222 .
   Google Scholar
• Preciso , L . 2000 . Regularity of the composition and of the inversion operator and perturbation analysis of the conformal sewing problem in Romieu type spaces . Tr. Inst. Mat., Minsk , 5 : 99 – 104 .
   Google Scholar