Boundary and Eigenvalue Problems

Abstract

Here some basic applications of variational methods are presented. We begin by proving the projection theorem for closed convex subsets of a Hilbert space $\mathcal{H}$ which is followed by the proof of the minimax principle for positive self-adjoint operators with discrete spectrum in a real Hilbert space. The next section explains the solution of the Dirichlet problem using the direct methods of the calculus of variations. As an illustration of the results on constrained minimization we prove the existence of eigenvalues for the Dirichlet Laplace operator. Next we explain how this strategy can be extended to general elliptic partial differential operators, first for the case of linear partial differential operators and then for certain classes of nonlinear partial differential operators. In each of these cases the existence of eigenvalues follows from the existence of a Lagrange multiplicator. In these applications we rely on some basic results about Sobolev spaces as explained in Part I. Finally we comment on some methods to determine critical points of functions which are not covered by our short introduction.