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Then there exist ˜ ∈ (ii) Let Ω Ω,r ]0, [, and an open neighborhood UγM in R2 of (γM , 1 − δ2,n ), and a n real analytic map UΩ˜r,1 of ] − Ω,r ˜ , Ω,r ˜ [×UγM to R , and real analytic ˜ Rn ) such that maps U r,j for j = 2, 3 of ] − ˜ , ˜ [×UγM to C 1,α (clΩ, ˜ Ω Ω,r Ω,r June 4, 2010 15:14 38 WSPC - Proceedings Trim Size: 9in x 6in GenProc-ISAAC-Final M. Dalla Riva and M. Lanza de Cristoforis (14) holds and such that u( , x) = δ2,n log r,1 U [ , ΞM,n [ ]] + U r,2 [ , ΞM,n [ ]](x) γ( ) Ω˜ γ( ) Ω˜ n−1 + γ( ) UΩ˜r,3 [ , ΞM,n [ ]](x) ˜ , ∀ ∈]0, ∀x ∈ clΩ ˜ [.

A. Plamenewskii, 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. , 111, 112, Birkh¨ auser Verlag, Basel, 2000. 11. S. Ozawa, Electrostatic capacity and eigenvalues of the Laplacian, J. Fac. Sci. Univ. Tokyo Sect. , 30, (1983), 53–62. 12. J. B. Keller, Strong localized perturbations of eigenvalue problems, SIAM J. Appl. , 53, (1993), 770–798. V. com In multidimensional case we give an extension of the Lindel¨ of-Lehto-Virtanen theorem for normal functions and Lindel¨ of-Gehring-Lohwater theorem involving two paths for bounded functions.

8, Applied Mathematics and Mechanics, Vol. 8 (Academic Press, New York, 1964). 7. V. Kozlov, V. Maz’ya and A. Movchan, Asymptotic analysis of fields in multi- June 4, 2010 15:14 30 8. 9. 10. 11. WSPC - Proceedings Trim Size: 9in x 6in GenProc-ISAAC-Final M. Dalla Riva structures, Oxford Mathematical Monographs (The Clarendon Press Oxford University Press, New York, 1999). V. Maz’ya, S. Nazarov and B. Plamenevskij, Asymptotic theory of elliptic boundary value problems in singularly perturbed domains.

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