On positive solutions of Liouville-Gelfand problem

Abstract

Modern science is highly interested in processes in nonlinear media. Mathematical models of these processes are often described by boundary-value problems for nonlinear elliptic equations. And the construction of two-sided approximations to the desired function is a perspective direction of solving such problems.The purpose of this work is to consider the existence and uniqueness of a regular positive solution to the Liouville-Gelfand problem and justify the possibility of constructing two-sided approximations to a solution. The two-sided approximations monotonically approximate the desired solution from above and below, and therefore have such an important advantage over other approximate methods that they provide an opportunity to obtain a convenient a posteriori estimate of the error of the calculations.The study of the Liouville-Gelfand problem is carried out by methods of the operator equations theory in partially ordered spaces. The mathematical model of the problem under consideration is the Dirichlet problem for a nonlinear elliptic equation with a positive parameter. The established properties of the corresponding nonlinear operator equation have given us an opportunity to obtain a condition for an input parameter, which guarantees the existence and uniqueness of the regular positive solution, as well as the possibility of constructing two-sided approximations, regardless of the domain geometry in which the problem is considered. The corresponding Liouville-Gelfand problem of the operator equation contains the Green's function for the Laplace operator of the first boundary value problem, and therefore the condition that the input parameter satisfies also contains it. Since the Green's function is known for a small number of relatively simple domains, Green's quasifunction method is used to solve the problem in domains of complex geometry. We note that the Green's quasifunction can be constructed practically for a domain of any geometry.The proposed approach allows us: a) to obtain a formula, which the parameter in the problem statement must satisfy, regardless of the domain geometry; b) for the first time, construct two-sided approximations to a solution to the Liouville-Gelfand problem; c) for the first time to obtain an a priori estimate of the solution depending on the selected value of the parameter in the problem statement.The proposed method has advantages over other approximate methods in relative simplicity of the algorithm implementation. The proposed method can be used for solving applied problems with mathematical models that are described by boundary value problems for nonlinear elliptic equations. In cases when the Green's function is unknown or has a complex form, the application of the Green's quasifunction method is proposed.

References

[1] D. A. Frank-Kamenetskii Diffusion and heat exchange in chemical kinetics (Princeton: Princeton Univ. Press, 1955), 382.
[2] Ya. B. Zel’dovich “K teorii rasprostranenija plameni [Theory of flame propagation]”, Zhurnal fizicheskoj himii. Vol. 22, No. 1 (1948) : 27-48.
[3] T. Aubin Some nonlinear problems in Rie-mannian geometry (Berlin: Springer-Verlag, 1998), 398. doi: 10.1007/978-3-662-13006-3.
[4] C. Bandle Isoperimetric inequalities and applications (London: Pitman, 1980), 228.
[5] B. Gidas, W. Ni and L. Nirenberg “Symmetry and related properties via the maximum principle” Comm. Math. Phys. Vol. 68, No. 3 (1979) : 209-243.
[6] J. Liouville “Sur l’equation aux derivees partielles d2 log =dudv ± 2a2 = 0” J. Math. Pures Appl. Vol. 18 (1853) : 71-72.
[7] G. Bratu “Sur les equations integrales non lineaires” Bulletin de la Societe Mathematique de France. Vol. 42 (1914) : 113-142. doi: 10.24033/bsmf.943.
[8] S. Chandrasekhar An introduction to the study of stellar structure (New York: Dover Pub., Inc., 1957), 509.
[9] I. M. Gelfand “Some problems in the theory of quasilinear equations” American Mathematical Society Translations: Series 2. Vol. 29 (1963) : 295-381. doi: 10.1090/trans2/029.
[10] D. Joseph and T. Lundgren “Quasilinear Dirichlet problems driven by positive sources” Arch. Rat. Mech. Anal. Vol. 49, No. 4 (1973) : 241–269. doi: 10.1007/BF00250508.
[11] D. Ye and F. Zhou “A generalized two dimensional Emden-Fowler equation with exponential nonlinearity”, Calculus of Variations and Partial Differential Equations. Vol. 13, No. 2 (2001) : 141-158. doi: 10.1007/s005260000069.
[12] Y. Bozhkov “Noether Symmetries and Critical Exponents” Symmetry, Integrability and Geometry: Methods and Applications. Vol. 1, No. 022 (2005) : 1-12. doi: 10.3842/SIGMA.2005.022.
[13] V. L. Rvachev, A. P. Slesarenko and N. A. Safonov “Matematicheskoe modelirovanie teplovogo samovosplamenenija dlja stacionarnyh uslovij metodom R-funkcij [Mathematical modeling of thermal autoignition for stationary conditions using R-functions method]” Doklady AN Ukrainy. Serija A. No. 12 (1992) : 24-27.
[14] N. S. Kurpel Projection-iterative methods for solution of operator equations (Providence: American Mathematical Society, 1976), 196.
[15] M. A. Krasnoselskii Positive solutions of operator equations (Groningen: P. Noordhoff, 1964), 381.
[16] V. I. Opoitsev “A generalization of the theory of monotone and concave operators” Trans. Moscow, Math. Soc. Vol. 2 (1979) : 243-279.
[17] V. I. Opoitsev and T. A. Khurodze Nelinejnye operatory v prostranstvah s konusom [Nonlinear operators in spaces with a cone] (Tbilisi: Izd-vo Tbilisskogo un-ta, 1984), 270.
[18] A. I. Kolosov “Ob odnom klasse uravnenij s vognutymi operatorami, zavisjashhimi ot parametra [A class of equations with concave operators that depend on a parameter]” Matematicheskie zametki. Vol. 49, No. 4 (1991) : 74-80.
[19] V. L Rvachev Teorija R-funkcij i nekotorye ee prilozhenija [Theory of R-functions and some applications] (Kiev: Nauk. dumka, 1982), 552.
[20] S. V. Kolosova, M. V. Sidorov “Primenenie iteracionnyh metodov k resheniju jellipticheskih kraevyh zadach s jeksponencial’noj nelinejnost’ju [Application of iterative methods to the solution of elliptic boundary value problems with exponential nonlinearity]” Radiojelektronika i informatika. No. 3 (62) (2013) : 28-31.
[21] S. V. Kolosova, V. S. Lukhanin, M. V. Sidorov “O nekotoryh podhodah k resheniju kraevyh zadach dlja nelinejnyh jellipticheskih uravnenij [On some approaches to the solution of boundary value problems for nonlinear elliptic equations]”Trudy XVI Mezhdunarodnogo simpoziuma «Metody diskretnyh osobennostej v zadachah matematicheskoj fiziki» (MDOZMF-2013). (2013) : 205-208.
[22] S. V. Kolosova, V. S. Lukhanin “Pro dodatni rozv’jazki odniyeyi zadachi z geterotonnim operatorom ta pro pobudovu poslidovnih nablizhen’ [On positive solutions of one problem with heterotone operator and the construction of successive approximations]” Visnik Harkivs’kogo nacional’nogo universitetu imeni V.N. Karazina. Serija Matematichne modeljuvannja. Informacijni tehnologiyi. Avtomatizovani sistemi upravlinnja. No. 31 (2016) : 59-72.
[23] I. V. Svirsky Metody tipa Bubnova-Galerkina i posledovatel’nyh priblizhenij [Methods of the Bubnov-Galerkin type and a sequence of approximation] (Moscow: Nauka, 1968), 199.
Published
2018-12-22
How to Cite
KOLOSOVA, S. V.; LUKHANIN, V. S.; SIDOROV, M. V.. On positive solutions of Liouville-Gelfand problem. Journal of Mathematics, Mechanics and Computer Science, [S.l.], v. 99, n. 3, p. 78-91, dec. 2018. ISSN 1563-0277. Available at: <http://bm.kaznu.kz/index.php/kaznu/article/view/460>. Date accessed: 21 jan. 2019.
Keywords Green’s function, Green’s quasifunction, two-sided approximations, invariant cone segment, monotone operator