Weak Formulations of Elliptic Boundary Value Problems

Abstract

In this chapter, we consider weak formulations of some elliptic boundary value problems and study the well-posedness of the variational problems. We begin with a derivation of a weak formulation of the homogeneous Dirichlet boundary value problem for the Poisson equation. In the abstract form, a weak formulation can be viewed as an operator equation. In the second section, we provide some general results on existence and uniqueness for linear operator equations. In the third section, we present and discuss the well-known Lax-Milgram Lemma, which is applied, in the section following, to the study of well-posedness of variational formulations for various linear elliptic boundary value problems. We also apply the Lax-Milgram Lemma in studying a boundary value problem in linearized elasticity; this is done in Section 8.5. The framework in the Lax-Milgram Lemma is suitable for the development of the Galerkin method for numerically solving linear elliptic boundary value problems. In Section 8.6, we provide a brief discussion of two different weak formulations: the mixed formulation and the dual formulation. For the development of Petrov-Galerkin method, where the trial function space and the test function space are different, we discuss a generalization of Lax-Milgram Lemma in Section 8.7. Most of the chapter is concerned with boundary value problems with linear differential operators. In the last section, we analyze a nonlinear elliptic boundary value problem.