Variational Formulations of Elliptic Boundary Value Problems

  • Kendall Atkinson
  • Weimin Han
Part of the Texts in Applied Mathematics book series (TAM, volume 39)


In this chapter, we formulate variational (or weak) forms of some elliptic boundary value problems and study the well-posedness of the variational problems. We begin with a derivation of the 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. 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 7.6, 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 7.7. 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.


Bilinear Form Variational Formulation Weak Formulation Elliptic Boundary Mixed Formulation 
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Copyright information

© Springer-Verlag New York, Inc. 2001

Authors and Affiliations

  • Kendall Atkinson
    • 1
  • Weimin Han
    • 2
    • 3
  1. 1.Department of Mathematics, Department of Computer ScienceUniversity of IowaIowa CityUSA
  2. 2.Department of MathematicsUniversity of IowaIowa CityUSA
  3. 3.Department of MathematicsZhejiang UniversityHangzhouPeople’s Republic of China

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