Abstract
The structure of this chapter can be described as follows. Sections 5.1 and 5.2 give several results for asymptotic approximations of Green’s kernels in domains with singularly perturbed smooth or conical exterior boundaries. Section 5.3 presents a detailed analysis of Green’s function of the Dirichlet–Neumann problem in a long cylindrical body. We introduce the notion of a capacitary potential and its asymptotic approximation in the elongated domain and construct an asymptotic approximation of Green’s function in the long rod in Sects. 5.3.2 and 5.3.3.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
G. Allaire, Homogenization and two-scale convergence. SIAM J. Math. Anal. 23, 1482–1518 (1992)
N.S. Bakhvalov, G.P. Panasenko, Homogenization: Averaging Processes in Periodic Media (Kluwer, Dordrecht, 1989), (translated from Russian: Osrednenie Processov v Periodicheskih Sredah (Nauka, Moscow, 1984))
R.W. Barnard, K. Pearce, C. Campbell, A survey of applications of the Julia variation. Ann. Univ. Mariae Curie-Sklodowska Sect. A 54, 1–20 (2000)
G.A. Chechkin, Homogenization in perforated domains. In Topics on Concentration Phenomena and Problems with Multiple Scales, ed. by A. Braides, V. Chiadó Piat. Lecture Notes of the Unione Matematica Italiana, vol. 2 (Springer, Berlin, 2006), pp. 189–208
D. Cioranescu, F. Murat, A Strange Term Brought from Somewhere Else, Nonlinear Partial Differential Equations and their Applications, Collège de France Seminar, vol. II and III. Research Notes in Mathematics, vol. 60 and 70, pp. 98–138 and 154–178 (1982)
R. Courant, D. Hilbert, Methods of Mathematical Physics, vol. I (Interscience, New York, 1953)
M. Englis, D. Lukkassen, J. Peetre, L.E. Persson, On the formula of Jacques-Louis Lions for reproducing kernels of harmonic and other functions. J. Reine Angew. Math. 570, 89–129 (2004)
G. Fichera, Il teorema del massimo modulo per l’equazione dell’elastostatica tridimensionale. Arch. Ration. Mech. Anal. 7(1), 373–387 (1961)
R. Figari, A. Teta, A boundary value problem of mixed type on perforated domains. Asymptotic Anal. 6, 271–284 (1993)
R. Figari, G. Papanicolaou, J. Rubinstein, The point interaction approximation for diffusion in regions with many small holes. Stochastic Methods in Biology. Lect. Note Biomater. 70, 75–86 (1987)
G. Fremiot, J. Sokolowski, Shape Sensitivity Analysis of Problems with Singularities. Shape Optimization and Optimal Design (Cambridge University Press, Cambridge 1999), pp. 255–276, Lecture Notes in Pure and Appl. Math., vol. 216 (Dekker, New York, 2001)
D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order (Springer, Berlin, 1983)
J. Hadamard, Sur le problème d’analyse relatif à l’équilibre des plaques élastiques encastrées. Mémoire couronnéen 1907 par l’Académie: Prix Vaillant, Mémoires présentés par divers savants à l’Académie des Sciences 33, No 4 (1908). In Oeuvres de Jacques Hadamard, vol. 2 (Centre National de la Recherche Scientifique, Paris, 1968), pp. 515–629
A. Hönig, B. Niethammer, F. Otto, On first-order corrections to the LSW theory I: Infinite systems. J. Stat. Phys. 119, 61–122 (2005)
A.M. Il’in, Matching of Asymptotic Expansions of Solutions of Boundary Value Problems, Translations of Mathematical Monographs (American Mathematical Society, Providence, 1992)
G. Julia, Sur une équation aux dérivées fonctionnelles analogue à l’équation de M. Hadamard. C.R. Acad. Sci. Paris 172, 831–833 (1921)
G. Komatsu, Hadamard’s variational formula for the Bergman kernel. Proc. Jpn. Acad. Ser. A. Math. Sci. 58(8), 345–348 (1982)
V.A. Kondratiev, O.A. Oleinik: On the behavior at infinity of solutions of elliptic systems with a finite energy integral. Arch. Ration. Mech. Anal. 99(1), 75–89 (1987)
V. Kozlov, V. Maz’ya, A. Movchan, Fields in Multi-Structures, Asymptotic Analysis (Oxford University Press, Oxford, 1999)
N.S. Landkof, Foundations of Modern Potential Theory (Spinger, Berlin, 1972)
V.A. Marchenko, E.Y. Khruslov, Homogenization of Partial Differential Equations (Birkhäuser, Boston, 2006). Russian Edition: Kraevye zadachi v oblastiakh s melkozernistoi granitsei (Naukova dumka, Kiev, 1974)
V. Maz’ya, Sobolev Spaces (Springer, Berlin, 1985)
V.G. Maz’ya, A.B. Movchan, Uniform asymptotic formulae for Green’s kernels in regularly and singularly perturbed domains. C. R. Acad. Sci. Paris. Ser. I 343, 185–190 (2006)
V.G. Maz’ya, A.B. Movchan, Uniform asymptotic formulae for Green’s functions in singularly perturbed domains. J. Comput. Appl. Math. 208(1), 194–206 (2007)
V. Maz’ya, A. Movchan, Uniform asymptotic approximations of Green’s functions in a long rod. Math. Meth. Appl. Sci. 31, 2055–2068 (2008)
V. Maz’ya, A. Movchan, Uniform asymptotics of Green’s kernels for mixed and Neumann problems in domains with small holes and inclusions, in Sobolev Spaces in Mathematics III. Applications in Mathematical Physics (Springer, New York, 2009), pp. 277–316
V. Maz’ya, A. Movchan, Asymptotic treatment of perforated domains without homogenization. Math. Nachr. 283(1), 104–125 (2010)
V. Maz’ya, A. Movchan, M. Nieves, Green’s kernels for transmission problems in bodies with small inclusions, Operator Theory and Its Applications, In Memory of V. B. Lidskii (1924–2008), ed. by M. Levitin, D. Vassiliev, American Mathematical Society Translations, Series 2, vol. 231 (American Mathematical Society, Providence, RI, 2010), pp. 127–171
V. Maz’ya, A. Movchan, M. Nieves, Mesoscale asymptotic approximations to solutions of mixed boundary value problems in perforated domains. Multiscale Model. Simulat. 9(1), 424–448 (2011)
V. Maz’ya, S. Nazarov, B. Plamenevskii, Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains, vols. 1–2 (Birkhäuser, Boston, 2000)
V. Maz’ya, B. Plamenevskii, Estimates in L p and in Hölder classes and the Miranda-Agmon maximum principle for solutions of elliptic boundary value problems in domains with singular points on the boundary. In Elliptic boundary value problems ed. by V.Maz’ya et al. AMS Translations, Series 2, v. 123, 1–56 (1984)
V.G. Maz’ya, A.B. Movchan, M.J. Nieves, Uniform asymptotic formulae for Green’s tensors in elastic singularly perturbed domains with multiple inclusions, Rendiconti della accademia nazionale delle scienze detta dei XL, Memorie di matematica e applicazioni, Serie V, Vol. XXX, Parte I, 103–158 (2006)
V.G. Maz’ya, A.B. Movchan, M.J. Nieves, Uniform asymptotic formulae for Green’s tensors in elastic singularly perturbed domains. Asymptotic Anal. 52(3/4), 173–206 (2007)
A.B. Movchan, N.V. Movchan, C.G. Poulton, Asymptotic Models of Fields in Dilute and Densely Packed Composites (Imperial College Press, London, 2002)
O.A. Oleinik, G.A. Yosifian, Boundary value problems for second order elliptic equations in unbounded domains and Saint-Venant’s principle. Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 4e série, tome 4(2), 269–290 (1977)
S. Ozawa, Approximation of Green’s function in a region with many obstacles. In Geometry and Analysis on Manifolds, ed. by T. Sunada, Lecture Notes in Mathematics, vol. 1339 (Springer, New York, 1988), pp. 212–225
B. Palmerio, A. Dervieux, Hadamard’s variational formula for a mixed problem and an application to a problem related to a Signorini-like variational inequality. Numer. Funct. Anal. Optim 1(2), 113–144 (1979)
G. Pólya, G. Szegö, Isoperimetric Inequalities in Mathematical Physics (Princeton University Press, Princeton, 1951)
E. Sánchez-Palencia, Non-homogeneous Media and Vibration Theory. Lecture Notes in Physics, vol. 27 (Springer, New York, 1980)
E. Sánchez-Palencia, Homogenization Method for the Study of Composite Media. Asymptotic Analysis, II. Lecture Notes in Math., vol. 985 (Springer, Berlin, 1983), pp. 192–214
E. Sánchez-Palencia, Homogenization in mechanics. A survey of solved and open problems. Rend. Sem. Mat. Univ. Politec. Torino 44(1), 1–45 (1986)
E.M. Stein, Singular Integrals and Differentiability Properties of Functions (Princeton University Press, Princeton, 1970)
V.V. Zhikov, S.M. Kozlov, O.A. Oleinik, Homogenization of Differential Operators (Nauka, Moscow, 1993); English transl., Homogenization of Differential Operators and Integral Functionals (Springer, Berlin, 1994)
V.V. Zhikov, Averaging of problems in the theory of elasticity on singular structures. Izv. Ross. Akad. Nauk Ser. Mat. 66(2), 81–148 (2002); English transl., Izv. Math. 66(2), 299–365 (2002)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Maz’ya, V., Movchan, A., Nieves, M. (2013). Other Examples of Asymptotic Approximations of Green’s Functions in Singularly Perturbed Domains. In: Green's Kernels and Meso-Scale Approximations in Perforated Domains. Lecture Notes in Mathematics, vol 2077. Springer, Heidelberg. https://doi.org/10.1007/978-3-319-00357-3_5
Download citation
DOI: https://doi.org/10.1007/978-3-319-00357-3_5
Published:
Publisher Name: Springer, Heidelberg
Print ISBN: 978-3-319-00356-6
Online ISBN: 978-3-319-00357-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)