Journal of Applied Mathematics and Decision Sciences
Volume 4 (2000), Issue 2, Pages 193-204

Variational methods for boundary value problems

B. Tsang,1,2 S. W. Taylor,1 and G. C. Wake2

1Department of Mathematics, University of Auckland, Auckland, New Zealand
2Department of Mathematics and Statistics, University of Canterbury, Christchurch, New Zealand

Copyright © 2000 B. Tsang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The variational formulation of boundary value problems is valuable in providing remarkably easy computational algorithms as well as an alternative framework with which to prove existence results. Boundary conditions impose constraints which can be annoying from a computational point of view. The question is then posed: what is the most general boundary value problem which can be posed in variational form with the boundary conditions appearing naturally? Special cases of two-point problems in one-dimension and some higher dimensional problems are addressed. There is a deep connection with self-adjointness for the linear case. Further cases under which a Lagrangian may or may not exist are explained.