Goa, MengistuWodajeneh, Terefe2019-05-062023-11-042019-05-062023-11-042018-09-05http://etd.aau.edu.et/handle/123456789/18193Partial differential equations arise naturally as models for many physical phenomena. The unknown function u then describes the state of a physical system (for example, the temperature distribution or the shape of a soap film realizing the least surface area amongst all surfaces spanned by a wire) and the given function E describes the physical laws according to which the state evolves or behaves (possibly also including interaction with external forces). In this thesis elliptic boundary problems of the form { are analyzed. The main focus is obtaining the weak formulation of the problem, what is called the variational formation. The weak form of solution to the problem is obtained by applying the celebrated Lax-Milgram Theorem which is the generalization of Reisz Representation Theorem to the variational formulation of the problem.enAnalysis of Elliptic PartialDifferential EquaionsSobolev SpaceElliptic BoundaryThesis