Geodesic
Solvability of the Boundary-Value Problem
Matematičeskie zametki, Tome 80 (2006) no. 1, pp. 60-68.

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

The solvability of the boundary-value problem for a string-beam model is studied. The model is described by an equation of orders 2 and 4 on different edges of an arbitrary graph. Criteria for the problem to be degenerate and nondegenerate are obtained; in particular, it is proved that the nondegeneracy of the problem is equivalent to the maximum principle.
Keywords: geometric graph (network), ordinary differential equation on a graph, boundary-value problem, nondegeneracy, degeneracy, maximum principle. 
@article{MZM_2006_80_1_a7,
     author = {K. P. Lazarev and T. V. Beloglazova},
     title = {Solvability of the {Boundary-Value} {Problem}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {60--68},
     publisher = {mathdoc},
     volume = {80},
     number = {1},
     year = {2006},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2006_80_1_a7/}
}
TY  - JOUR
AU  - K. P. Lazarev
AU  - T. V. Beloglazova
TI  - Solvability of the Boundary-Value Problem
JO  - Matematičeskie zametki
PY  - 2006
SP  - 60
EP  - 68
VL  - 80
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2006_80_1_a7/
LA  - ru
ID  - MZM_2006_80_1_a7
ER  - 
%0 Journal Article
%A K. P. Lazarev
%A T. V. Beloglazova
%T Solvability of the Boundary-Value Problem
%J Matematičeskie zametki
%D 2006
%P 60-68
%V 80
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2006_80_1_a7/
%G ru
%F MZM_2006_80_1_a7
K. P. Lazarev; T. V. Beloglazova. Solvability of the Boundary-Value Problem. Matematičeskie zametki, Tome 80 (2006) no. 1, pp. 60-68. http://geodesic.mathdoc.fr/item/MZM_2006_80_1_a7/

[1] Yu. V. Pokornyi, O. M. Penkin, V. L. Pryadiev, A. V. Borovskikh, K. P. Lazarev, S. A. Shabrov, Differentsialnye uravneniya na geometricheskikh grafakh, Fizmatlit, M., 2003

[2] P. Kuchment, “Graph models for waves in then structures”, Waves Random Media, 12 (2002), 1–24 | DOI | MR

[3] Yu. V. Pokornyi, “O neostsillyatsii obyknovennykh differentsialnykh uravnenii i neravenstv na prostranstvennykh setyakh”, Differents. uravneniya, 37:5 (2001), 661–671 | MR | Zbl

[4] O. M. Penkin, Yu. V. Pokornyi, E. N. Provotorova, “Ob odnoi vektornoi kraevoi zadache”, Kraevye zadachi, Perm, 1983, 64–70 | MR | Zbl

[5] O. M. Penkin, Yu. V. Pokornyi, “O kraevoi zadache na grafe”, Differents. uravneniya, 24:4 (1988), 701–703 | MR | Zbl

[6] Yu. V. Pokornyi, O. M. Penkin, “Teoremy Shturma dlya uravnenii na grafakh”, Dokl. AN SSSR, 309:6 (1989), 1306–1308 | MR

[7] Yu. V. Pokornyi, O. M. Penkin, “O teoremakh sravneniya dlya uravnenii na grafakh”, Differents. uravneniya, 25:7 (1989), 1141–1150 | MR | Zbl

[8] Yu. V. Pokornyi, E. N. Provotorova, O. M. Penkin, “O spektre nekotorykh vektornykh kraevykh zadach”, Voprosy kachestvennoi teorii differentsialnykh uravnenii, Sb. nauchn. tr., Nauka, Novosibirsk, 1988, 109–113 | MR

[9] J. E. Lagnese, G. Leugering, E. J. P. G. Schmidt, “Control of planar networks of Timoshenko beams”, SIAM J. Control Optim., 31 (1993), 780–811 | DOI | MR | Zbl

[10] B. Dekoninck, S. Nicaise, “Spectre des réseaux de poutres”, C.R. Acad. Sci. Paris, 326. Série 1 (1998), 1249–1254 | MR | Zbl

[11] B. Dekoninck, S. Nicaise, “The eigenvalue problem for networks of beams”, Generalized Functions. Operator Theory and Dynamical Systems, Chapman and Hall Research in Math., 1999, 335–344 | MR | Zbl

[12] F. Ali Mehmeti, B. Dekoninck, “Transient vibrations of planar networks of beams: interaction of flexion, transversal and longitudal waves”, Partial Differential Equations on Multistructures, Lect. Notes Pure Appl. Math., 219, eds. F. Ali Mehmeti, J. von Below and S. Nicaise, 2001, 1–18 | MR | Zbl

[13] B. Dekoninck, “Control of network of Euler–Bernoulli beams”, ESAIM-COCV, 4 (1999), 57–82 | DOI | MR

[14] J. E. Lagnese, G. Leugering, E. J. P. G. Schmidt, “Modeling of dynamic networks of thin thermoelastic beams”, Math. Meth. Appl. Sci., 16 (1993), 327–358 | DOI | MR | Zbl

[15] A. V. Borovskikh, K. P. Lazarev, “Fourth-order differential equations on geometric graphs”, J. Math. Sci., 119:6 (2004), 719–739 | DOI | MR

[16] A. V. Borovskikh, R. Mustafokulov, K. P. Lazarev, Yu. V. Pokornyi, “Ob odnom klasse differentsialnykh uravnenii chetvertogo poryadka na prostranstvennoi seti”, Dokl. RAN, 345:6 (1995), 730–732 | MR | Zbl

[17] Yu. V. Pokornyi, R. Mustafokulov, “O pozitivnoi obratimosti nekotorykh kraevykh zadach dlya uravnenii chetvertogo poryadka”, Differents. uravneniya, 33:10 (1997), 1358–1365 | MR | Zbl

[18] Yu. V. Pokornyi, R. Mustafokulov, “O polozhitelnosti funktsii Grina lineinykh kraevykh zadach dlya uravnenii chetvertogo poryadka na grafe”, Izv. vuzov. Matematika, 1999, no. 2, 75–82 | MR | Zbl

[19] Yu. V. Pokornyi, K. P. Lazarev, “Nekotorye ostsillyatsionnye teoremy dlya mnogotochechnykh zadach”, Differents. uravneniya, 23:4 (1987), 658–670 | MR | Zbl

[20] K. P. Lazarev, O spektre nekotorykh negladkikh mnogotochechnykh zadach, Diss. ... k.f.-m.n., VGU, Voronezh, 1988

[21] E. N. Provotorova, “O vektornykh kraevykh zadachakh, porozhdaemykh skalyarnym differentsialnym operatorom”, Differents. uravneniya, 23:10 (1987), 1711–1715 | MR

[22] Yu. V. Pokornyi, T. V. Beloglazova, K. P. Lazarev, “Ob odnom klasse raznoporyadkovykh obyknovennykh differentsialnykh uravnenii na grafe”, Matem. zametki, 73:3 (2003), 469–470 | MR | Zbl