Sobolev Spaces and Variational Method Applied to Elliptic Partial Differential Equations

Abstract

Variational methods have proved to be very important in the study of optimal shape, time, velocity, volume or energy. Laws existing in mechanics, physics, astronomy, economics and other fields of natural sciences and engineering obey variational principles. The main objective of variational method is to obtain the solutions governed by these prin- ciples. Fermat postulated that light follows a part of least possible time, this is a subject in finding minimizers of a given functional. It is important to note that we are in this work concerned about solution of some Boudary Value Problem of some Partial Differential Equa- tions. The Boundary Value Problem is formulated in abstract form as; A(u) = 0 in Ω, B(u) = 0 on ∂Ω, Ω ⊂ RN open, (1) where A(u) = 0 denotes a given Partial Differential Equation for unknown u and B(u) = 0 is a given boundary value condition. The problem of interest in the variational method shall be existence and the regularity of the minimizers of an associated functional. In chapter three, we discussed Optimization in infinite dimensional spaces, a topic which is very important in the study of variational methods. We specifically studied the application of the variational methods in solving the Dirichlet Homogeneous Boundary Value Problem:   −∆u = f in Ω,  u = 0 on ∂Ω, where Ω is bounded open subset of RN of class C 1 and f ∈ L2 (Ω). It is important to understand the meaning of Linear Elliptic Partial Differential Equations, since our work is targeted towards a method of solving such Partial Differential Equations. Definition A partial differential equation(PDE) is an equation involving partial derivatives of an unknown function u : Ω → R, where Ω is an open subset of Rn , n ≥ 2. The order of a Partial Differential Equation is the order of the highest order derivative that appears in the Partial Differential Equation.