A periodic boundary value problem for a fourth order differential operator with a summable potential

S. I. Mitrokhin

Vladikavkazskij matematičeskij žurnal, Tome 19 (2017) no. 4, pp. 35-49

Résumé

The paper is devoted to the study of a fourth-order differential operator with a summable potential and periodic boundary conditions. The method of studying of operators with a summable potential is an extension of the method of studying operators with piecewise smooth coefficients. Boundary value problems of this kind arise when studying the oscillations of beams and bridges composed from materials of different density. The solution of the differential equation is reduced to the solution of the Volterra integral equation. The integral equation is solved by Picard's method of successive approximations. The aim of the, investigation of the integral equation is to obtain asymptotic formulas and estimates for the solutions of the differential equation that defines the differential operator. Questions of geophysics, quantum mechanics, kinetics, gas dynamics and the theory of oscillations of rods, beams and membranes require the development of asymptotic methods for the case of differential equations with nonsmooth coefficients. Asymptotic methods continue to evolve, despite the rapid progress in numerical methods associated with the advent of supercomputers; at present asymptotic and numerical methods complement each other. In the paper, for large values of the spectral parameter, the asymptotics of the solutions of the differential equation that defines the differential operator is obtained. Asymptotic estimates for solutions are established similarly to the asymptotic estimates of solutions of a second-order differential operator with smooth coefficients. The study of periodic boundary conditions leads to the study of the roots of a function represented in the form of a fourth order determinant. To obtain the roots of this function, an indicator diagram has been examined. The roots are in four sectors of an infinitesimal angle, determined by the indicator diagram. The behavior of the roots of this equation in each of the sectors of the indicator diagram is investigated. The asymptotics of eigenvalues of the differential operator under consideration is found. The formulas obtained for the asymptotics of the eigenvalues make it possible to study the spectral properties of the eigenfunctions. If the potential of the operator is not a summable function, but only piecewise smooth, then the obtained formulas for the asymptotics of the eigenvalues are sufficient to derive the formula for the first regularized trace of the differential operator under study.

Bibliographie

[1] Lidskyi V. B., Sadovnichiy V. A., “Asymptotic formulas for the roots of a class of entire functions”, Mathematical Collection, 65:4 (1968), 558–566 (in Russian)

[2] Lidskyi V. B., Sadovnichiy V. A., “Regularized sums of the roots of a class of entire functions”, Functional Analysis and its Applications, 1:2 (1967), 52–59 (in Russian)

[3] Mitrokhin S. I., “About formulas of regularized traces for second order differential operators with discontinuous coefficients”, Vestnik MGU. Series: Mathematics, Mechanics, 1986, no. 6, 3–6 (in Russian)

[4] Mitrokhin S. I., “About spectral properties of differential operators with discontinuous coefficients”, Differential Equations, 28:3 (1992), 530–532 (in Russian) | Zbl

[5] Mitrokhin S. I., “About some spectral properties of differential operators of the second order with discontinuous weight function”, Reports of the Russian Academy of Sciences, 356:1 (1997), 13–15 (in Russian)

[6] Sadovnichiy V. A., “About traces of ordinary differential operators of the highest orders”, Mathematical Collection, 72:2 (1967), 293–310 (in Russian)

[7] Vinokurov V. A., Sadovnichii V. A., “Asymptotics of any order for eigenvalues and eigenfunctions of the boundary value Sturm–Liouville problem on a segment with a summable potential”, Differential Equations, 34:10 (1998), 1423–1426 (in Russian) | Zbl

[8] Vinokurov V. A., Sadovnichii V. A., “Asymptotics of any order for eigenvalues and eigenfunctions of the boundary value Sturm–Liouville problem on a segment with a summable potential”, News of the Russian Academy of Sciences, 64:4 (2000), 47–108 (in Russian) | DOI | Zbl

[9] Mitrokhin S. I., “About spectral properties of a fourth-order differential operator with integrable coefficients”, Works MIAN, 270 (2010), 188–197 (in Russian) | Zbl

[10] Mitrokhin S. I., “About spectral properties of a delay differential operator with summable coefficients”, Ufa Mathematical Journal, 3:4 (2011), 95–115 (in Russian)

[11] Mitrokhin S. I., “About spectral properties of differential operators of odd order with a summable potential”, Differential Equations, 47:2 (2011), 1808–1811 (in Russian) | Zbl

[12] Mitrokhin S. I., “Spectral properties of a family of differential operators of high even order with summable potential”, Vestnik VSU. Series: Physics. Mathematics, 2016, no. 4, 121–135 (in Russian)

[13] Naimark M. A., Linear Differential Operators, Nauka, M., 1969, 528 pp. (in Russian)

[14] Yurko V. A., Introduction to the Theory of Inverse Spectral Problems, Fizmatlit, M., 2007, 384 pp. (in Russian)

[15] Bellman R., Cooke K. L., Differential-Difference Equations, Mir, M., 1967, 548 pp. (in Russian)