Geodesic
Some aspects of the nontrivial solvability of homogeneous Dirichlet problems for linear equations of arbitrary even order in the disk
Matematičeskie zametki, Tome 77 (2005) no. 4, pp. 498-508.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper, we obtain a necessary and sufficient condition for the nontrivial solvability of homogeneous Dirichlet problems in the disk for linear equations of arbitrary even order $2m$ with constant complex coefficients and homogeneous nondegenerate symbol in general position. The cases $m=1,2,3$ are studied separately. For the case $m=2$, we consider examples of real elliptic systems reducible to single equations with constant complex coefficients for which the homogeneous Dirichlet problem in the disk has a countable set of linearly independent polynomial solutions. 
@article{MZM_2005_77_4_a1,
     author = {V. P. Burskii and E. A. Buryachenko},
     title = {Some aspects of the nontrivial solvability of homogeneous {Dirichlet} problems for linear equations of arbitrary even order in the disk},
     journal = {Matemati\v{c}eskie zametki},
     pages = {498--508},
     publisher = {mathdoc},
     volume = {77},
     number = {4},
     year = {2005},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2005_77_4_a1/}
}
TY  - JOUR
AU  - V. P. Burskii
AU  - E. A. Buryachenko
TI  - Some aspects of the nontrivial solvability of homogeneous Dirichlet problems for linear equations of arbitrary even order in the disk
JO  - Matematičeskie zametki
PY  - 2005
SP  - 498
EP  - 508
VL  - 77
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2005_77_4_a1/
LA  - ru
ID  - MZM_2005_77_4_a1
ER  - 
%0 Journal Article
%A V. P. Burskii
%A E. A. Buryachenko
%T Some aspects of the nontrivial solvability of homogeneous Dirichlet problems for linear equations of arbitrary even order in the disk
%J Matematičeskie zametki
%D 2005
%P 498-508
%V 77
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2005_77_4_a1/
%G ru
%F MZM_2005_77_4_a1
V. P. Burskii; E. A. Buryachenko. Some aspects of the nontrivial solvability of homogeneous Dirichlet problems for linear equations of arbitrary even order in the disk. Matematičeskie zametki, Tome 77 (2005) no. 4, pp. 498-508. http://geodesic.mathdoc.fr/item/MZM_2005_77_4_a1/

[1] Bitsadze A. V., “O edinstvennosti resheniya zadachi Dirikhle dlya ellipticheskikh uravnenii s chastnymi proizvodnymi”, UMN, 3:6 (1948), 211–212 | Zbl

[2] Bitsadze A. V., Uravneniya smeshannogo tipa, Fizmatgiz, M., 1959

[3] Lions Zh.-L., Madzhenes E., Neodnorodnye granichnye zadachi i ikh prilozheniya, T. 1, Mir, M., 1971

[4] Markovskii A. I., “Ob oblasti znacheniya ellipticheskogo polinoma”, Ukr. matem. zh., 25:2 (1973), 228–234 | MR | Zbl

[5] Vishik M. I., “O silno ellipticheskikh sistemakh differentsialnykh uravnenii”, Matem. sb., 29:3 (1951), 615–676 | Zbl

[6] Aleksandryan R. A., “O zadache Dirikhle dlya uravneniya struny i o polnote odnoi sistemy funktsii v kruge”, Dokl. AN SSSR, 73:5 (1950)

[7] Aleksandryan R. A., “Spektralnye svoistva operatorov, porozhdennykh sistemami differentsialnykh uravnenii tipa S. L. Soboleva”, Tr. MMO, 9, URSS, M., 1960, 455–507

[8] Burskii V. P., “O narushenii edinstvennosti resheniya zadachi Dirikhle dlya ellipticheskikh sistem v kruge”, Matem. zametki, 48:3 (1990), 32–36

[9] Babayan A. O., “On unique solvability of Dirichlet problem for fourth order properly elliptic equation”, Izv. Nat. AN Armenii. Matem., 34:5 (1999), 5–18 | Zbl

[10] Buryachenko E. A., “O edinstvennosti reshenii zadachi Dirikhle v kruge dlya differentsialnykh uravnenii chetvertogo poryadka v vyrozhdennykh sluchayakh”, Nelineinye granichnye zadachi, 10 (2000), 44–49 | Zbl

[11] Burskii V. P., “O edinstvennosti resheniya nekotorykh granichnykh zadach dlya differentsialnykh uravnenii v oblasti s algebraicheskoi granitsei”, Ukr. matem. zh., 45:7 (1993), 898–906 | Zbl

[12] Vekua I. N., Novye metody resheniya ellipticheskikh uravnenii, OGIZ, M., 1948

[13] Bitsadze A. V., Nekotorye klassy differentsialnykh uravnenii v chastnykh proizvodnykh, Mir, M., 1981

[14] Beitmen G., Erdeii A., Vysshie transtsendentnye funktsii, T. 2, Nauka, M., 1974

[15] Buryachenko E. A., “K voprosu o narushenii edinstvennosti resheniya zadachi Dirikhle dlya uravnenii s chastnymi proizvodnymi chetvertogo poryadka”, Tr. IPMM NAN Ukrainy, 4 (1999), 15–21 | Zbl