The Eshelby theorem and the problem on optimal patch

S. A. Nazarov

Algebra i analiz, Tome 21 (2009) no. 5, pp. 155-195.

Résumé

Let $\Omega^0$ be an ellipsoidal inclusion in the Euclidean space ${\mathbb{R}}^n$. It is checked that if a solution of the homogeneous transmission problem for a formally selfadjoint elliptic system of second order differential equations with piecewise smooth coefficients grows linearly at infinity, then this solution is a linear vector-valued function in the interior of $\Omega^0$. This fact generalizes the classical Eshelby theorem in elasticity theory and makes it possible to indicate simple and explicit formulas for the polarization matrix of the inclusion in the composite space, as well as to solve a problem about optimal patching of an elliptical hole.

Bibliographie

[1] Nečas J., Les méthodes directes en théorie des équations elliptiques, Masson et Cie, Paris; Academia, Prague, 1967 | MR | Zbl

[2] Nazarov S. A., “Samosopryazhennye ellipticheskie kraevye zadachi. Polinomialnoe svoistvo i formalno polozhitelnye operatory”, Probl. mat. anal., 16, SPbGU, SPb., 1997, 167–192

[3] Nazarov S. A., “Polinomialnoe svoistvo samosopryazhennykh ellipticheskikh kraevykh zadach i algebraicheskoe opisanie ikh atributov”, Uspekhi mat. nauk, 54:5 (1999), 77–142 | MR | Zbl

[4] Lekhnitskii S. G., Teoriya uprugosti anizotropnogo tela, Nauka, M., 1977 | MR | Zbl

[5] Nazarov S. A., Asimptoticheskaya teoriya tonkikh plastin i sterzhnei. Ponizhenie razmernosti i integralnye otsenki, Nauch. kn., Novosibirsk, 2002 | Zbl

[6] Polia G., Sege G., Izoperimetricheskie neravenstva v matematicheskoi fizike, Fizmatgiz, M., 1962

[7] Zorin I. S., Movchan A. B., Nazarov S. A., “O primenenii tenzorov uprugoi emkosti, polyarizatsii i prisoedinennoi deformatsii”, Issled. po uprugosti i plastichnosti, 16, LGU, L., 1990, 75–91 | MR

[8] Nazarov S. A., “Tenzor i mery povrezhdennosti. I. Asimptoticheskii analiz anizotropnoi sredy s defektami”, Izv. RAN. Mekh. tverd. tela, 2000, no. 3, 113–124

[9] Nazarov S. A., “Asimptoticheskie usloviya v tochkakh, samosopryazhennye rasshireniya operatorov i metod sraschivaemykh razlozhenii”, Tr. S.-Peterburg. mat. o-va, 5, 1998, 112–183 | Zbl

[10] Movchan A. B., “Matritsy polyarizatsii i emkosti Vinera dlya operatora teorii uprugosti v dvusvyaznykh oblastyakh”, Mat. zametki, 47:2 (1990), 151–153 | MR | Zbl

[11] Movchan A. B., “Integral characteristics of elastic inclusions and cavities in the two-dimensional theory of elesticity”, European J. Appl. Math., 3 (1992), 21–30 | DOI | MR | Zbl

[12] Movchan A. B., Movchan N. V., Mathematical modelling of solids with nonregular boundaries, CRC Press, Boca Raton, 1995 | MR

[13] Argatov I. I., “Integralnye kharakteristiki zhestkikh vklyuchenii i polostei v ploskoi teorii uprugosti”, Prikl. mat. i mekh., 62:2 (1998), 283–289 | MR | Zbl

[14] Eshelbi Dzh., Kontinualnaya teoriya dislokatsii, IL, M., 1963

[15] Kunin I. A., Elastic media with microstructure. II. Three dimensional models, Springer Ser. Solid-State Sci., 44, Springer-Verlag, Berlin, 1983 | MR | Zbl

[16] Kanaun S. K., Levin V. M., Metod effektivnogo polya v mekhanike kompozitnykh materialov, Petrozavod. un-t, Petrozavodsk, 1993 | MR | Zbl

[17] Kunin I. A., Sosnina E. G., “Ellipsoidalnaya neodnorodnost v uprugoi srede”, Dokl. AN SSSR, 199:3 (1971), 571–574 | Zbl

[18] Kunin I. A., Sosnina E. G., “Kontsentratsiya napryazhenii na ellipsoidalnoi neodnorodnosti v anizotropnoi uprugoi srede”, Prikl. mat. i mekh., 37:2 (1973), 306–315 | Zbl

[19] Mura T., Micromechanics of defects in solids, 2nd ed., Kluwer, Dordrecht, 1987 | Zbl

[20] Freidin A. B., “On new phase inclusions in elastic solids”, ZAMM, 87:2 (2007), 102–116 | DOI | MR | Zbl

[21] Freidin A. B., Vilchevskaya E. N., “Multiple development of new phase inclusions in elastic solids”, Internat. J. Engrg. Sci., 47:2 (2009), 240–260 | DOI | MR

[22] Ladyzhenskaya O. A., Kraevye zadachi matematicheskoi fiziki, Nauka, M., 1973 | MR

[23] Lions Zh.-L., Madzhenes E., Neodnorodnye granichnye zadachi i ikh prilozheniya, Mir, M., 1971 | Zbl

[24] Kondratev V. A., “Kraevye zadachi dlya ellipticheskikh uravnenii v oblastyakh s konicheskimi ili uglovymi tochkami”, Tr. Mosk. mat. o-va, 16, 1967, 209–292

[25] Nazarov S. A., Plamenevskii B. A., Ellipticheskie zadachi v oblastyakh s kusochno gladkoi granitsei, Nauka, M., 1991

[26] Pazy A., “Asymptotic expansions of solutions of ordinary differential equations in Hilbert space”, Arch. Rational Mech. Anal., 24 (1967), 193–218 | DOI | MR | Zbl

[27] Gelfand I. M., Shilov G. E., Obobschennye funktsii i deistviya nad nimi, Fizmatgiz, M., 1959

[28] Mazya V. G., Plamenevskii B. A., “O koeffitsientakh v asimptotike reshenii ellipticheskikh kraevykh zadach v oblastyakh s konicheskimi tochkami”, Math. Nachr., 76 (1977), 29–60 | DOI

[29] Mazya V. G., Plamenevskii B. A., “Otsenki v $L_p$ i v klassakh Gëldera i printsip maksimuma Miranda–Agmona dlya reshenii ellipticheskikh kraevykh zadach v oblastyakh s osobymi tochkami na granitse”, Math. Nachr., 81 (1978), 25–82 | DOI

[30] Lobo M., Nazarov S. A., Perez E., “Eigen-oscillations of contrasting non-homogeneous elastic bodies: asymptotic and uniform estimates for eigenvalues”, IMA J. Appl. Math., 70:3 (2005), 419–458 | DOI | MR | Zbl

[31] Roǐtberg Ya. A., Elliptic boundary value problems in the spaces of distributions, Math. Appl., 384, Kluwer Acad. Publ. Group, Dordrecht, 1996 | MR

[32] Mikhlin S. G., Mnogomernye singulyarnye integraly i integralnye uravneniya, Fizmatgiz, M., 1962 | MR | Zbl

[33] Agmon S., Douglis A., Nirenberg L., “Estimates near the boundary for soluitons of elliptic partial differential equations satisfying general boundary conditions. II”, Comm. Pure Appl. Math., 17:1 (1964), 35–92 | DOI | MR | Zbl

[34] Roitberg Ya. A., Sheftel Z. G., “Obschie granichnye zadachi dlya ellipticheskikh uravnenii s razryvnymi koeffitsientami”, Dokl. AN SSSR, 148:5 (1963), 1034–1037 | MR

[35] Ilin A. M., Soglasovanie asimptoticheskikh razlozhenii reshenii kraevykh zadach, Nauka, M., 1989 | MR

[36] Mazja W. G., Nasarow S. A., Plamenewski B. A., Asymptotische Theorie elliptischer Randwertaufgaben in singulär gestörten Gebieten, I, Math. Lehrbücher Monogr. II. Abt. Math. Monogr., 82, Akademie-Verlag, Berlin, 1991 ; Maz'ya V., Nazarov S., Plamenevskij B., Asymptotic theory of elliptic boundary value problems in singularly perturbed domains, Vol. 1, Oper. Theory Adv. Appl., 111, Birkhäuser-Verlag, Basel, 2000 | MR | MR

[37] Mazya V. G., Nazarov S. A., “Asimptotika integralov energii pri malykh vozmuscheniyakh granitsy vblizi uglovykh i konicheskikh tochek”, Tr. Mosk. mat. o-va, 50, 1987, 79–129 | MR

[38] Nazarov S. A., Shpekovius-Noigebauer M., “Approksimatsiya neogranichennykh oblastei ogranichennymi. Kraevye zadachi dlya operatora Lame”, Algebra i analiz, 8:5 (1996), 229–268 | MR | Zbl

[39] Nazarov S. A., Sokolowski J., “Asymptotic analysis of shape functionals”, J. Math. Pures Appl. (9), 82:2 (2003), 125–196 | MR | Zbl

[40] Nazarov S. A., Specovius-Neugebauer M., “Approximation of exterior problems. Optimal conditions for the Laplacian”, Analysis, 16:4 (1996), 305–324 | MR | Zbl

[41] Nazarov S. A., Sokolowski J., “Self-adjoint extensions for the Neumann Laplacian and applications”, Acta Math. Sin. (Engl. Ser.), 22:3 (2006), 879–906 | DOI | MR | Zbl

[42] Langer S., Nazarov S. A., Shpekovius-Noigebauer M., “Affinnye preobrazovaniya trekhmernykh anizotropnykh sred i yavnye formuly dlya fundamentalnykh matrits”, Prikl. mekh. i tekhn. fiz., 47:2 (2006), 95–102 | MR | Zbl

[43] Kupradze V. D., Gegeliya T. G., Basheleishvili M. O., Burchuladze T. V., Trekhmernye zadachi matematicheskoi teorii upurugosti i termouprugosti, Nauka, M., 1976 | MR

[44] Lifshits I. M., Rozentsveig L. N., “O postroenii tenzora Grina dlya osnovnogo uravneniya teorii uprugosti v sluchae neogranichennoi uprugo-anizotropnoi sredy”, Zh. eksperim. i teor. fiz., 17:9 (1947), 783–791 | MR

[45] Kröner E., “Das Fundamentalintegral der anisotropen elastischen Differentialgleichungen”, Z. Physik, 136 (1953), 402–410 | DOI | MR | Zbl