Вестник КазНУ. Серия математика, механика, информатика (Dec 2018)

On positive solutions of Liouville-Gelfand problem

  • S. V. Kolosova,
  • V. S. Lukhanin,
  • M. V. Sidorov

Journal volume & issue
Vol. 99, no. 3
pp. 78 – 91


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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.