On the index and spectrum of integral operators of potential type along Radon curves

V. Yu. Shelepov

Sbornik. Mathematics, Tome 70 (1991) no. 1, pp. 175-203

Résumé

A study is made of how classical integral equations of mathematical physics are affected by nonregularity of the contour of integration. A criterion is obtained for a matrix integral equation with operator of potential type acting in $L_p$ $(1<p<\infty)$ to be Noetherian, and the index is computed. It is established that an integral equation corresponding to the interior Dirichlet problem for harmonic functions is Noetherian in $L_p$ for all $p$ except for a finite or countable number of values determined by the angles of the contour; the defect numbers, which depend on $p$ and the angles mentioned, are found. Analogous results are obtained for the system of integral equations of the planar theory of elasticity. The non-Noetherian spectrum of a matrix integral operator of potential type acting in a space of continuous vector-valued functions is described. This result is illustrated by an example of an operator in elasticity theory (for which, in particular, the Fredholm radius is found) and of the direct value of a double layer potential.

Bibliographie

[1] Danilyuk I. I., Shelepov V. Yu., “Ob ogranichennosti v $L_p$ singulyarnogo operatora s yadrom Koshi vdol krivoi ogranichennogo vrascheniya”, DAN SSSR, 174:3 (1967), 514–517 | Zbl

[2] Danilyuk I. I., Shelepov V. Yu., “Pro obemezhenist u zvazhenikh prostorakh $L_p$ singulyarnikh integralnikh operatoriv vzdovzh linii obmezhenogo obertaniya”, Dokl. AN USSR, ser. A., 1969, no. 3, 199–202

[3] Shelepov V. Yu., “Ob indekse integralnogo operatora tipa potentsiala v prostranstve $L_p$”, DAN SSSR, 186:6 (1969), 1266–1268 | Zbl

[4] Bazalii B. V., Shelepov V. Yu., “O spektre potentsiala dvoinogo sloya na krivoi ogranichennogo vrascheniya”, Granichnye zadachi dlya differentsialnykh uravnenii, Naukova dumka, Kiev, 1980, 13–30 | MR

[5] Shelepov V. Yu., “Ob issledovanii metodom Ya. B. Lopatinskogo matrichnykh integralnykh uravnenii v prostranstve nepreryvnykh funktsii”, Obschaya teoriya granichnykh zadach, Naukova dumka, Kiev, 1983, 220–226 | MR

[6] Shelepov V. Yu., “Neterovost integralnykh uravnenii ploskoi teorii uprugosti v prostranstvakh $L_p$, $p>1$”, Dokl. AN USSR, ser. A., 1988, no. 8, 27–31 | MR | Zbl

[7] Radon I., “O lineinykh preobrazovaniyakh i funktsionalnykh uravneniyakh”, UMN, 1936, no. 1, 200–227

[8] Nikolskii S. M., “Lineinye uravneniya v lineinykh normirovannykh prostranstvakh”, Izv. AN SSSR. Ser. matem., 7:3 (1943), 147–166 | MR | Zbl

[9] Gokhberg I. Ts., Krein M. G., “Sistemy integralnykh uravnenii na polupryamoi s yadrami, zavisyaschimi ot raznosti argumentov”, UMN, 13:2 (1958), 3–72 | MR

[10] Goluzin G. M., Geometricheskaya teoriya funktsii kompleksnogo peremennogo, Nauka, M., 1966 | MR

[11] Radon I., “O kraevykh zadachakh dlya logarifmicheskogo potentsiala”, UMN, 1:3–4(13–14) (1946), 96–124 | MR | Zbl

[12] Danilyuk I. I., Neregulyarnye granichnye zadachi na ploskosti, Nauka, M., 1975 | MR

[13] Lopatinskii Ya. B., “Pro odin tip singulyarnikh integralnikh rivnyan”, Teoretichna i prikladna matematika, no. 2, Lvovsk. gos. un-t, 1963, 53–57 | MR

[14] Zigmund A., Trigonometricheskie ryady, t. 2, Mir, M., 1965 | MR

[15] Kantorovich L. V., Akilov G. P., Funktsionalnyi analiz v normirovannykh prostranstvakh, Fizmatgiz, M., 1959 | MR

[16] Muskhelishvili N. I., Nekotorye osnovnye zadachi matematicheskoi teorii uprugosti, Nauka, M., 1966

[17] Parasyuk E. N., “Ob indekse integralnogo operatora, sootvetstvuyuschego vtoroi osnovnoi zadache ploskoi teorii uprugosti”, Ukr. matem. zhurn., 16:2 (1964), 250–254

[18] Gradshtein I. S., Ryzhik I. M., Tablitsy integralov, summ, ryadov i proizvedenii, Fizmatgiz, M., 1962

[19] Muskhelishvili N. I., Singulyarnye integralnye uravneniya, Nauka, M., 1973 | MR

[20] Burago Yu. D., Mazya V. G., “Nekotorye voprosy teorii potentsiala i teorii funktsii dlya oblastei s neregulyarnymi granitsami”, Zap. nauch. seminarov LOMI, 3:1 (1967), 5–86 | MR

[21] Fabes E. B., Jodeit M., Lewis J. E., “Double Layer Potentials for Domains with Corners and Edges”, Indiana Univ. Math. J., 26:1 (1977), 95–114 | DOI | MR | Zbl

[22] Kresin G. I., Mazya V. G., “O suschestvennoi norme operatora tipa potentsiala dvoinogo sloya v prostranstve $C_m$”, DAN SSSR, 246:2 (1979), 272–275 | MR | Zbl

[23] Kral J., Integral Operators in Potential Theory, Lect. Notes in Math., 823, Springer Verlag, Berlin etc., 1980 | MR | Zbl

[24] Mikhailov S. E., “Ob integralnom uravnenii nekotorykh kraevykh zadach dlya garmonicheskikh funktsii v ploskikh mnogosvyaznykh oblastyakh s neregulyarnoi granitsei”, Matem. sb., 121(163) (1983), 533–544

[25] Verchota G., “Layer Potentials and Regularity for the Dirichlet Problem for Laplace's Equation in Lipschitz Domains”, J. Functional Analysis, 59 (1984), 572–611 | DOI | MR | Zbl

[26] Kenig C., “Recent Progress on Boundary-Value Problems on Lipschitz Domains”, Poc. of Symp. in Pure Math. Amer. Math. Soc., 43 (1985), 175–205 | MR | Zbl

[27] Mazya V. G., “K teorii potentsiala dlya sistemy Lame v oblasti s kusochno-gladkoi granitsei”, Differentsialnye uravneniya i ikh primeneniya, Tbilisi, 1986, 123–129 | MR

[28] Grisvard P., “Le probléme de Dirichlet pour les équations de Lamé”, C. r. Acad. sci, sér. 1, 304:3 (1987), 71–73 | MR | Zbl

[29] Maslennikova V. N., Bogovskii M. E., “K $L_p$-teorii ellipticheskikh kraevykh zadach dlya oblastei s negladkimi i nekompaktnymi granitsami”, Funktsionalnye i chislennye metody matem. fiziki, Naukova dumka, Kiev, 1988, 142–150 | MR