Abstract
In contrast with the well-known methods of matching asymptotics and multiscale (or compound) asymptotics, the “functional analytic approach” of Lanza de Cristoforis (Analysis (Munich) 28:63–93, 2008) allows to prove convergence of expansions around interior small holes of size \(\varepsilon \) for solutions of elliptic boundary value problems. Using the method of layer potentials, the asymptotic behavior of the solution as \(\varepsilon \) tends to zero is described not only by asymptotic series in powers of \(\varepsilon \), but by convergent power series. Here we use this method to investigate the Dirichlet problem for the Laplace operator where holes are collapsing at a polygonal corner of opening \(\omega \). Then in addition to the scale \(\varepsilon \) there appears the scale \(\eta =\varepsilon ^{\pi /\omega }\). We prove that when \(\pi /\omega \) is irrational, the solution of the Dirichlet problem is given by convergent series in powers of these two small parameters. Due to interference of the two scales, this convergence is obtained, in full generality, by grouping together integer powers of the two scales that are very close to each other. Nevertheless, there exists a dense subset of openings \(\omega \) (characterized by Diophantine approximation properties), for which real analyticity in the two variables \(\varepsilon \) and \(\eta \) holds and the power series converge unconditionally. When \(\pi /\omega \) is rational, the series are unconditionally convergent, but contain terms in \(\log \varepsilon \).
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Ammari, H., Kang, H.: Polarization and Moment Tensors, Vol. 162 of Applied Mathematical Sciences. Springer, New York (2007). (With applications to inverse problems and effective medium theory)
Bergamasco, A.P.: Remarks about global analytic hypoellipticity. Trans. Am. Math. Soc. 351, 4113–4126 (1999)
Bonnaillie-Noël, V., Dalla Riva, M., Dambrine, M., Musolino, P.: A Dirichlet problem for the Laplace operator in a domain with a small hole close to the boundary. Preprint. arXiv:1612.04637 (2016)
Brahimi, M., Dauge, M.: Analyticité et problèmes aux limites dans un polygone. C. R. Acad. Sci. Paris Sér. I Math. 294, 9–12 (1982)
Costabel, M.: Boundary integral operators on Lipschitz domains: elementary results. SIAM J. Math. Anal. 19, 613–626 (1988)
Costabel, M., Dauge, M.: Construction of corner singularities for Agmon–Douglis–Nirenberg elliptic systems. Math. Nachr. 162, 209–237 (1993)
Costabel, M., Dauge, M.: Stable asymptotics for elliptic systems on plane domains with corners. Commun. Partial Differ. Equ. 9–10, 1677–1726 (1994)
Costabel, M., Wendland, W.L.: Strong ellipticity of boundary integral operators. J. Reine Angew. Math. 372, 34–63 (1986)
Dalla Riva, M., Musolino, P.: Real analytic families of harmonic functions in a planar domain with a small hole. J. Math. Anal. Appl. 422, 37–55 (2015)
Dalla Riva, M., Musolino, P.: A mixed problem for the Laplace operator in a domain with moderately close holes. Commun. Partial Differ. Equ. 41, 812–837 (2016)
Dauge, M.: Second membre analytique pour un problème aux limites elliptique d’ordre \(2m\) sur un polygone. Commun. Partial Differ. Equ. 9, 169–195 (1984)
Dauge, M.: Elliptic Boundary Value Problems on Corner Domains—Smoothness and Asymptotics of Solutions, Vol. 1341 of Lecture Notes in Mathematics. Springer, Berlin (1988)
Dauge, M., Tordeux, S., Vial, G.: Selfsimilar perturbation near a corner: matching versus multiscale expansions for a model problem. Around the Research of Vladimir Maz’ya. II, Vol. 12 of Int. Math. Ser. (N. Y.), pp. 95–134. Springer, New York (2010)
Folland, G.B.: Introduction to Partial Differential Equations, 2nd edn. Princeton University Press, Princeton (1995)
Greenfield, S.J., Wallach, N.R.: Global hypoellipticity and Liouville numbers. Proc. Am. Math. Soc. 31, 112–114 (1972)
Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers, 6th edn. Oxford University Press, Oxford (2008). (Revised by D. R. Heath-Brown and J. H. Silverman, With a foreword by Andrew Wiles)
Himonas, A.A.: Global analytic and Gevrey hypoellipticity of sublaplacians under Diophantine conditions. Proc. Am. Math. Soc. 129, 2061–2067 (2001)
Il’in, A.M.: Matching of asymptotic expansions of solutions of boundary value problems, vol. 102 of Translations of Mathematical Monographs, American Mathematical Society, Providence 1992. Translated from the Russian by V. Minachin [V. V. Minakhin]
Kondrat’ev, V.A.: Boundary-value problems for elliptic equations in domains with conical or angular points. Trans. Moscow Math. Soc. 16, 227–313 (1967)
Kozlov, V., Maz’ya, V., Movchan, A.: Asymptotic Analysis of Fields in Multi-structures, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, Oxford Science Publications (1999)
Lanza de Cristoforis, M.: Asymptotic behaviour of the conformal representation of a Jordan domain with a small hole in Schauder spaces. Comput. Methods Funct. Theory 2, 1–27 (2002)
Lanza de Cristoforis, M.: Asymptotic behavior 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 (2008)
Lanza de Cristoforis, M., Musolino, P.: A real analyticity result for a nonlinear integral operator. J. Integral Equ. Appl. 25, 21–46 (2013)
Mayboroda, S., Mitrea, M.: Layer potentials and boundary value problems for Laplacian in Lipschitz domains with data in quasi-Banach Besov spaces. Ann. Mat. Pura Appl. (4) 185, 155–187 (2006)
Maz’ya, V., Nazarov, S., Plamenevskij, B.: Asymptotic theory of elliptic boundary value problems in singularly perturbed domains. Vols. I, II, vol. 111, 112 of Operator Theory: Advances and Applications. Birkhäuser, Basel (2000)
McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge (2000)
Morrey Jr., C.B., Nirenberg, L.: On the analyticity of the solutions of linear elliptic systems of partial differential equations. Commun. Pure Appl. Math. 10, 271–290 (1957)
Steinbach, O., Wendland, W.L.: On C. Neumann’s method for second-order elliptic systems in domains with non-smooth boundaries. J. Math. Anal. Appl. 262, 733–748 (2001)
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Costabel, M., Dalla Riva, M., Dauge, M. et al. Converging Expansions for Lipschitz Self-Similar Perforations of a Plane Sector. Integr. Equ. Oper. Theory 88, 401–449 (2017). https://doi.org/10.1007/s00020-017-2377-7
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DOI: https://doi.org/10.1007/s00020-017-2377-7