This seminar report basically consists of two chapters. In the first part, the Sobolevrnand Nikol Skii Besov functions spaces are investigated and some inclusion theoremsrnare developed as well as some relations of the two function spaces are stated as arnparticular case. As the main objective of the seminar report we formulate thernnecessary and sufficient condition for the function (which is equal to the solution ofrnthe Dirichlet problem on the boundary) on the boundary of some region, for thernsolution of the generalized formulation of the Dirichlet problem of Laplace equatonrnto exist. For the classical formulation of the Dirichlet problem, we, as an assumption,rntake the value of the function on the boundary of some region in which the solutionrnis defined is continuous on the boundary. But, this does not guarantee the exis tencernof the solution of the problem, which is shown by Hadamard's example. Therefore,rnas a conclusion, we will see that the formulation of the necessary and sufficientrncondition for the generalized formulation of the Dirichlet problem of Laplacernequaton in the second chapter of this semina r report.rnBefore all, T thank the almighty God with out His help nothing is happened.rnI would like to express my special gratitude to my advisor and teacher Dr. TsegayernGedif for his unlimited help for the completion of this seminar report. T am reallyrnlucky for his being here with me. I would also like to thank all my teachers for theirrncon tribution to bring me to this level.rnFinally, I thank my parents for all they have given me and for their prayers. Also,rnI would like to thank W /0 Tobiaw Tefera, for typing this manuscript partially.