Weak Convergence Methods for Energy Minimization

This compact set of notes present some basic, abstract results with two examples of boundary-value problems on weak convergence methods for the study of existence of minimizers of essentially convex functionals. The method described here is called the direct method in the calculus of variations. The main results are stated in their simplest possible setting. But, they have many variations that can be applied to more complicated problems. The two examples are: (1) A standard Dirichlet type integral whose minimizers are solutions to the Poisson equation; (2) The elastic energy functional of displacement field whose minimizers are solutions to the fundamental equations in the linear theory of elasticity. The definition and some useful properties of the space H(Ω), H 0 (Ω), and H (Ω) are given in Appendix (cf. Section A). Some of the properties are proved for special cases.