Variational Methods In Partial Differential Equations

This thesis is a summary on variational methods, whose origin and developmentand application in partial differential equations will be introduced.Calculous, which was founded mainly by Newton, is the greatest achievement in mathematics in 17th century. In the late 17th century, mathematicians (they are also physicists in that time) had found some new mathematical problems in solving the physical problems by calculus, such as maximum and minimum of integrals. The first variational problem was introduced and solved by Newton . In his great work-Mathematical Principles in Natural Philosophy, he studied the shape of rotational surface precessing by a constant velocity along the axis and with minimal resistance. In 18th century, Euler and Lagrange's work led to the foundation of a new mathematical branch--variational methods, which is used to solve the maximum or minimum problems.Let us recall the form of problems which introduce the mathematicians into variational methods. Consider the functionalJ(y)=∫_x1^x2 f{x,y(x),y'(x))dx,where f is a continuous function, y is a differentiable function with respect to x. Seek a function y(x) from (x₁,y₁) to (x₂,y₂) such that J achieve its minimum. Euler found the essential method to dealing with those kinds of problems. It was shown that the minimal function must solve the following equationf_y(x,y(x),y'(x)) -d/dx f_y'(x,y(x),y'(x)) = 0,which is the so-called Euler-Lagrange equation.The basic content of variational methods is to seek extremal points and critical points, which can be transformed from or into boundary problems of partial differential equations. That is to say, one can solve a boundary problem of a partial differential equation by solving a corresponding variational problem. We give an example. Assume that H is an Hilbert space, A is a symmetric positive-definite operator in D_a (?) H, and f∈H. Then, u∈D_A is a solution ofAu = f (1)if and only if u is a solution ofI(u) = (?) I(v), (2)whereI(v) = 1/2(Av,v)-(f,v), v∈D_A (3)is a quadratic functional. It is shown from this result that solving the operatorequation (1) is equivalent to solving the variational problem (2). Here, the equation (1) is the Euler-Lagrange equation of the quadratic functional (3).In the latter two sections of this thesis, we will introduce the application of variational methods in linear and nonlinear partial differential equations, respectively. We will state the application of variational methods in linear partial differentialequations by taking Poisson equation as a typical example. Assume thatΩ(?)Rⁿ is a bounded domain and v is the unit external normal vector on (?)Ω. For Poisson equation-â–³u(x) = f(x), x∈Ω,we will consider three kinds of boundary problems with the following boundary value conditions, respectively,1. the first boundary condition (Dirichlet condition) : u|(?)Ω= 0;2. the second boundary condition (Neumann condition) : (?)u/(?)v|_(?)Ω= 0;3. the third boundary condition (Robin condition) : (?)u/(?)v+α(x)u|_(?)Ω= 0 withα(x)≥α₀≥0. For the Hilbert space H = L²(Ω),-â–³is just the positive-definite operator onrespectively. The above three boundary problems of Poisson equation can be transformed into the variational problems of seeking the minimum points of the functionalson the corresponding sets D₁, D_θ, D_σ, respectively. By proving the existence of the minimum points, we prove the existence of the solutions of the boundary problems. Additionally, we also prove the solvability of the eigenvalue problem of Laplace equation by using the Lagrange multiplier method.We introduce the application of the variational methods in nonlinear partial differential equations by considering the following Dirichlet problem of a semilinearelliptic equation-â–³u(x) + f(x,u(x)) = 0, x∈Ω, (4)u(x) = 0, x∈(?)Ω, (5)whereΩ(?) Rⁿ is a bounded domain and f satisfies some structural conditions. Generally speaking, we are perhaps not able to solve the nonlinear problem by using variational methods to seek the minimum points of the corresponding variationalproblem since it may even be unbounded. For this reason, we will solve the nonlinear problem by seeking the critical points of the corresponding variational problem. The Euler-Lagrange equation of the problem is justI(v) = 1/2∫_Ω|â–½v(x)|²dx -∫_ΩF(x,v(x))dx, v∈H₀¹(Ω), (6)whereF(x,v)=∫₀^v f(x,t)dt.The functional (6) is unbounded, so the extremal point may not exist. By using the mountain pass lemma and the Lagrange multiplier method, we prove the existence of the critical point of the functional (6). Thus the problem (4), (5) is solvable. Finally, for an operator equation with general form, we prove the existence of the critical point of the corresponding functional by the mountain pass lemma, and thus the equation is solvable. This result can be applied to solve the semilinear wave equation.