Abstract
This chapter discusses the Dirichlet problem and the Neumann problem for a half-space. The two main types of boundary-value problem for Laplace's equation are the Dirichlet problem and the Neumann problem. The Dirichlet problem for Laplace's equation consists in determining a function u in the domain Ω with the boundary S to satisfy the equation 2u = 0, and the Neumann problem consists in finding a solution of the equation mentioned above to satisfy the boundary conditions: δuδns = f2 (S).Taking as Ω the domain z > 0; the plane serving as the surface S, the chapter reviews that in such a domain, the solution of the Dirichlet problem, bounded everywhere, is unique; and that the solution of the Neumann problem is determined to within an additive constant.
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