Eigenvalue Problems of Second Order Linear Elliptic Operators

Eigenvalue problems have a wide range of applications. In particular, the existence of positive solutions to second order semi-linear and quasi-linear elliptic equations and systems depends critically on the principal eigenvalue (the first or smallest eigenvalue) of a corresponding eigenvalue problem. In this chapter, we introduce the theory of eigenvalue problems for second order linear elliptic operators. These results will be used extensively in the later chapters. In the last chapter, we will also introduce the eigenvalue problem for the p-Laplace operator.