Geodesic
Dynamic plane deformation of semi-infinite polygonal plate: Kostrov's “paradox” and its amendment
Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 54, Tome 533 (2024), pp. 170-185 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

Under dynamic loading of a concave isotropic wedge the usual formulas which express the displacement field through two potentials and applies for any convex wedge, lead to a strong singularity at the vertex and need to be improved (so-called Kostrov's correction). For an unbounded isotropic and homogeneous plane polygonal body, we derive a construction of the potentials providing true singularities of the displacement field in vertices of several “entering” corners. We also correct inaccuracies found in previous publications. 
@article{ZNSL_2024_533_a10,
     author = {S. A. Nazarov},
     title = {Dynamic plane deformation of semi-infinite polygonal plate: {Kostrov's} {\textquotedblleft}paradox{\textquotedblright} and its amendment},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {170--185},
     year = {2024},
     volume = {533},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2024_533_a10/}
}
TY  - JOUR
AU  - S. A. Nazarov
TI  - Dynamic plane deformation of semi-infinite polygonal plate: Kostrov's “paradox” and its amendment
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2024
SP  - 170
EP  - 185
VL  - 533
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2024_533_a10/
LA  - ru
ID  - ZNSL_2024_533_a10
ER  - 
%0 Journal Article
%A S. A. Nazarov
%T Dynamic plane deformation of semi-infinite polygonal plate: Kostrov's “paradox” and its amendment
%J Zapiski Nauchnykh Seminarov POMI
%D 2024
%P 170-185
%V 533
%U http://geodesic.mathdoc.fr/item/ZNSL_2024_533_a10/
%G ru
%F ZNSL_2024_533_a10
S. A. Nazarov. Dynamic plane deformation of semi-infinite polygonal plate: Kostrov's “paradox” and its amendment. Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 54, Tome 533 (2024), pp. 170-185. http://geodesic.mathdoc.fr/item/ZNSL_2024_533_a10/

[1] B. V. Kostrov, , “Difraktsiya ploskoi volny na zhestkom kline, vstavlennom bez treniya v bezgranichnuyu upruguyu sredu”, Prikladnaya matem. i mekhanika, 50:1 (1966), 198–202

[2] G. Dyuvo, Zh.-L. Lions, Neravenstva v mekhanike i fizike, M., 1980

[3] V. I. Smirnov, Kurs vysshei matematiki, chast 2, v. 4, Nauka, M., 1981 | MR

[4] G. Vatson, Teoriya besselevykh funktsii, IL, M., 1949

[5] O. M. Sapondzhyan, Izgib tonkikh uprugikh plit, Aiastan, Erevan, 1975

[6] M. Sh. Birman, “O variatsionnom metode Trefftsa dlya uravneniya $\Delta^2u=f$”, Dokl. AN SSSR, 101:2 (1955), 201–204 | Zbl

[7] S. G. Mikhlin, Variatsionnye metody v matematicheskoi fizike, Nauka, M., 1970 | MR

[8] M. Sh. Birman, G. E. Skvortsov, “O kvadratichnoi summiruemosti starshikh proizvodnykh resheniya zadachi Dirikhle v oblasti s kusochno gladkoi granitsei”, Izvestiya VUZ'ov. Matem., 1962, no. 5, 11–21

[9] V. A. Kondratev, “Kraevye zadachi dlya ellipticheskikh uravnenii v oblastyakh s konicheskimi ili uglovymi tochkami”, Trudy Moskovsk. matem. obschestva, 16, 1963, 219–292

[10] S. A. Nazarov, B. A. Plamenevsky, Elliptic problems in domains with piecewise smooth boundaries, Walter de Gruyter, Berlin–New York, 1994 | MR

[11] V. G. Mazya, S. A. Nazarov, “O paradokse Sapondzhyana–Babushki v zadachakh teorii tonkikh plastin”, Doklady AN ArmSSR, 78:3 (1984), 127–130 | Zbl

[12] S. A. Nazarov, G. Sweers, “A hinged plate equation and iterated Dirichlet Laplace operator on domains with concave corners”, J. of Differential Equations, 233:1 (2007), 151–180 | DOI | MR | Zbl

[13] M. Sh. Birman, M. Z. Solomyak, “$L^2$-teoriya operatora Maksvella v proizvolnykh oblastyakh”, Uspekhi matem. nauk, 42:6 (1987), 61–76 | MR | Zbl

[14] M. Sh. Birman, “Tri zadachi teorii sploshnykh sred v mnogogrannikakh”, Zap. nauchn. semin. POMI, 200, 1992, 27–37

[15] F. Assous, Jr. P. Ciarlet, E. Sonnendrücker, “Résolution des équations de Maxwell dans un domaine avec un coin rentrant”, C.R. Acad. Sci. Paris S ér. Math., 323:2 (1996), 203–208 | MR | Zbl

[16] M. Costabel, M. Dauge, “Singularities of electromagnetic fields in polyhedral domains”, Arch. Ration. Mech. Anal., 151:3 (2000), 221–276 | DOI | MR | Zbl

[17] M. Sh. Birman, “K teorii samosopryazhennykh rasshirenii polozhitelno opredelennykh operatorov”, Matem. sbornik, 38(80) (1956), 431–450 | Zbl

[18] F. A. Berezin, L. D. Faddeev, “Zamechanie ob uravnenii Shredingera s singulyarnym potentsialom”, Dokl. AN SSSR, 137:5 (1961), 1011–1014 | Zbl

[19] B. S. Pavlov, “Teoriya rasshirenii i yavno reshaemye modeli”, Uspekhi matem. nauk, 42:6 (1987), 99–132 | MR

[20] S. A. Nazarov, “Samosopryazhennye rasshireniya operatora zadachi Dirikhle v vesovykh funktsionalnykh prostranstvakh”, Matem. sbornik, 137:2 (1988), 224–241

[21] S. A. Nazarov, “Asimptoticheskie usloviya v tochkakh, samosopryazhennye rasshireniya operatorov i metod sraschivaemykh asimptoticheskikh razlozhenii”, Trudy Sankt-Peterburg. matem. o-va, 5 (1996), 112–183

[22] S. A. Nazarov, G. Kh. Svirs, “Kraevye zadachi dlya bigarmonicheskogo uravneniya i iterirovannogo laplasiana v trekhmernoi oblasti s rebrom”, Zap. nauchn. semin. POMI, 366, 2006, 153–198

[23] N. F. Morozov, I. L. Surovtsova, “Zadacha o dinamicheskom nagruzhenii ploskikh uprugikh oblastei s uglovymi tochkami kontura”, Prikladnaya matem. i mekhanika, 61:4 (1997), 654–659 | MR | Zbl

[24] L. Bers, F. John, M. Schehter, Partial differential equations, Interscience, New York, 1964 | MR | Zbl

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