Model of deformations of a stieltjes string system with a nonlinear condition

M. B. Zvereva

Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 32 (2022) no. 4, pp. 528-545

Résumé

In the present paper we study a model of deformations for a system of $n$ Stieltjes strings located along a geometric graph-star with a nonlinear condition at the node. The corresponding boundary value problem has the form $$ \left\{\begin{array}{lll} -\left(p_iu_i^\prime\right)(x)+\displaystyle{\int_0^x}u_idQ_i=F_i(x)-F_i(+0)-(p_iu_i')(+0), \quad i=1,2, \ldots, n,\\ \sum\limits_{i=1}^np_i(+0)u_i'(+0)\in N_{[-m,m]}u(0),\\u_1(0)=u_2(0)=\ldots=u_n(0)=u(0),\\(p_iu_i')(l_i-0)+u_i(l_i)\Delta Q_i(l_i)=\Delta F_i(l_i), \quad i=1,2,\ldots, n. \end{array} \right. $$ Here the functions $u_i(x)$ determine the deformations of each of the strings; $F_i(x)$ describe the distribution of the external load; $p_i(x)$ characterize the elasticity of strings; $Q_i(x)$ describe the elastic response of the environment. The jump $\Delta F_i(l_i)$ is equal to the external force concentrated at the point $l_i$; the jump $\Delta Q_i(l_i)$ coincides with the stiffness of the elastic support (spring) attached to the point $l_i$. The condition $\sum\limits_{i=1}^np_i(+0)u_i'(+0)\in N_{[-m,m]}u(0)$ arises due to the presence of a limiter in the node represented by the segment $ [-m,m]$, on the movement of strings under the influence of an external load, thus it is assumed that $|u(0)|\leq m$. Here $N_{[-m,m]}u(0)$ denotes the normal cone to $[-m,m]$ at the point $u(0)$. In the present paper a variational derivation of the model is carried out; existence and uniqueness theorems for solutions are proved; the critical loads at which the strings come into contact with the limiter are analyzed; an explicit formula for the representation of the solution is presented.

Détail

Keywords: Stieltjes integral, function of bounded variation, measure, geometric graph, energy functional.

Bibliographie

[1] Baranov V.N., Rodionov V.I., “O nelineinykh metricheskikh prostranstvakh funktsii ogranichennoi variatsii”, Vestnik Udmurtskogo universiteta. Matematika. Mekhanika. Kompyuternye nauki, 32:3 (2022), 341–360 | DOI | MR

[2] Burlutskaya M.Sh., Khromov A.P., “Operator Diraka s potentsialom spetsialnogo vida i periodicheskimi kraevymi usloviyami”, Differentsialnye uravneniya, 54:5 (2018), 592–601 | DOI

[3] Derr V.Ya., Kim I.G., “Prostranstvo pravilnykh funktsii i differentsialnoe uravnenie s obobschennymi funktsiyami v koeffitsientakh”, Vestnik Udmurtskogo universiteta. Matematika. Mekhanika. Kompyuternye nauki, 2014, no. 1, 3–18 | DOI

[4] Diab A.T., Kaldybekova B.K., Penkin O.M., “O kratnosti sobstvennykh znachenii v zadache Shturma-Liuvillya na grafakh”, Matematicheskie zametki, 99:4 (2016), 489–501 | DOI

[5] Kulaev R.Ch., “K voprosu ob ostsillyatsionnosti funktsii Grina razryvnoi kraevoi zadachi chetvertogo poryadka”, Matematicheskie zametki, 100:3 (2016), 375–387 | DOI

[6] Kulaev R.Ch., Urtaeva A.A., “Teoremy Shturma o raspredelenii nulei dlya uravneniya chetvertogo poryadka na grafe”, Matematicheskie zametki, 111:6 (2022), 947–952 | DOI

[7] Pokornyi Yu.V., Penkin O.M., Pryadiev V.L., Borovskikh A.V., Lazarev K.P., Shabrov S.A., Differentsialnye uravneniya na geometricheskikh grafakh, Fizmatlit, M., 2005

[8] Pokornyi Yu.V., Pryadiev V.L., “Nekotorye voprosy kachestvennoi teorii Shturma-Liuvillya na prostranstvennoi seti”, Uspekhi matematicheskikh nauk, 59:3 (357) (2004), 115–150 | DOI

[9] von Below J., Lubary J.A., Vasseur B., “Some remarks on the eigenvalue multiplicities of the Laplacian on infinite locally finite trees”, Results in Mathematics, 63 (2013), 1331–1350 | DOI | MR

[10] Burlutskaya M.Sh., “Metod Fure v smeshannoi zadache dlya volnovogo uravneniya na grafe”, Doklady Akademii nauk, 465:3 (2015), 519–522 | DOI

[11] Kamenskii M., Wen Ch.-F., Zvereva M., “On a variational problem for a model of a Stieltjes string with a backlash at the end”, Optimization, 69:9 (2020), 1935–1959 | DOI | MR

[12] Kramar Fijavž M., Mugnolo D., Nicaise S., “Dynamic transmission conditions for linear hyperbolic systems on networks”, Journal of Evolution Equations, 21:3 (2021), 3639–3673 | DOI | MR

[13] Pokorny Yu.V., Pryadiev V.L., “On conditions for transmission in the Sturm-Liouville problem on a network”, Journal of Mathematical Sciences, 130:5 (2005), 5013–5045 | DOI | MR

[14] Pokornyi Yu.V., Borovskikh A.V., “Differential equation on networks (geometric graphs)”, Journal of Mathematical Sciences, 119:6 (2004), 691–718 | DOI | MR

[15] Pokornyi Yu.V., Zvereva M.B., Bakhtina Zh.I., “Metod differentsiala Stiltesa v modelirovanii neregulyarnoi sistemy na geometricheskom grafe”, Differentsialnye uravneniya, 48:8 (2012), 1117–1125

[16] Pokornyi Yu.V., Zvereva M.B., Shabrov S.A., “Ostsillyatsionnaya teoriya Shturma-Liuvillya dlya impulsnykh zadach”, Uspekhi matematicheskikh nauk, 63:1 (379) (2008), 111–154 | DOI

[17] Provotorov V.V., Sergeev S.M., Part A.A., “Solvability of hyperbolic systems with distributed parameters on the graph in the weak formulation”, Vestnik Sankt-Peterburgskogo universiteta. Prikladnaya matematika. Informatika. Protsessy upravleniya, 15:1 (2019), 107–117 | DOI | MR

[18] Savchuk A.M., Shkalikov A.A., “Obratnye zadachi dlya operatora Shturma-Liuvillya s potentsialami iz prostranstv Soboleva. Ravnomernaya ustoichivost”, Funktsionalnyi analiz i ego prilozheniya, 44:4 (2010), 34–53 | DOI

[19] Vladykina V.E., Shkalikov A.A., “Asimptotika reshenii uravneniya Shturma-Liuvillya s singulyarnymi koeffitsientami”, Matematicheskie zametki, 98:6 (2015), 832–841 | DOI

[20] Yurko V.A., “Obratnye spektralnye zadachi dlya differentsialnykh operatorov na prostranstvennykh setyakh”, Uspekhi matematicheskikh nauk, 71:3 (429) (2016), 149–196 | DOI