Tsegaye GedifAli Eshetu2025-08-172025-08-172024-07https://etd.aau.edu.et/handle/123456789/6930The focus of this thesis is to examine weak or variational formulations of various elliptic boundary value problems and determine whether or not they are well-posed. The weak version of the homogeneous Dirichlet boundary value problem for the Poisson equation is first derived. A weak formulation can be thought of as an operator equation in its abstract form. Additionally, we offer some broad conclusions on the existence and uniqueness of linear operator equations.en-USVarational FormulationElliptic PartialDifferential EquationThesis