Applications of Functional Analysis to a Class of Elliptic PDES

Abstract

In this report we discuss about the eigensolutions of an eigenvalue problem for the Laplace operator by investigating the underlying variational problem. Mostly the discussion is restricted to case however it can be extended to other values of also. When we have constrained minimizers subject to norm, it is considered as the eigenvalue problem of Laplace operator. Several theorems are stated in the report which strongly support the existence of solutions to the variational problem, hence eigensolutions and also the sequence of eigensolutions. Solutions are treated as critical points (for the variational problem) in the sense of weak slope. Finally an additional necessary condition is introduced which is derived using inner variations to filter the non-eigensolutions of the problem. Basic concepts of functional analysis and necessary prerequisites to understand the variational problem and minimizers are also included.