On Discrete Elliptic Equation And Its Application In Mechanics

Steady-state solution of differential equation is useful in practice, so the properties of the steady-state solution are extensively considered after the mechanical models are established. In reality, many mechanical models are nonlinear elliptic equation with periodic boundary conditions, for instance, steady-state membrane vibration equation and steady-state flows in an ideal incompressible liquid on torus. Unfortunately, it's not easy to derive the exact solution for nonlinear differential equation, especially nonlinear partial differential equation. With the development of computer, advantage of finite difference method is showing. The original equation and condition are approximately replaced by algebraic system by using difference method. Solution of the system is the approximate solution of the original question. The first question of solving the algebraic system is the existence of the solution, so it's important to consider the existence of solutions for discrete elliptic equation with periodic boundary condition.In order to study the existence of solutions for discrete elliptic equation with periodic boundary condition further, we firstly introduce its application in mechanics and review the research on the nonlinear elliptic equation. After obtaining the eigenvalues of the corresponding eigen equation, some results are derived by using Mountain pass lemma, Linking theorem and some results in Morse theory. After that, a simple illustrative example and some remarks will be given. Results in this paper are useful not only in mechanics but also mathematics.