Variational Methods in Differential Equations
This chapter concerns classical variational methods in boundary value problems and a free boundary problem, with a special emphasis on how to view a differential equation as a variational problem. Variational methods are simple, but very powerful analytical tools for differential equations. In particular, the unique solvability of a differential equation reduces to a minimization problem, for which a minimizer is shown to be a solution to the original equation. As a model problem, the Poisson equation with different types of boundary conditions is considered. We begin with the derivation of the equation in the context of potential theory, and then show successful applications of variational methods to these boundary value problems. Finally, we study a free boundary problem by developing the idea to a minimization problem with a constraint.
KeywordsBoundary value problem Free boundary problem Variational method
- 1.D. Gilbarg, N.S. Trudinger, in Elliptic Partial Differential Equations of Second Order. Reprint of the 1998 edition. Classics in Mathematics (Springer, Berlin, 2001)Google Scholar