chapter  11
6 Pages

The Poisson Kernel

In this chapter we seek a solution u = u(x, t), x ∈ Rn, t > 0, of the Dirichlet problem for the Laplacian on the upper half space {(x, t) : x ∈ Rn, t > 0} given by {

∂2u ∂t2 (x, t) + (∆u)(x, t) = 0, x ∈ Rn, t > 0, u(x, 0) = f(x), x ∈ Rn, (11.1)

where f ∈ S. The problem in (11.1) is different in nature from the ones in Chapters 6 and 7. The t = 0 in Chapters 6 and 7 refers to the initial time and t = 0 in this chapter is understood to be the boundary of the upper half space {(x, t) : x ∈ Rn, t > 0}. The problem in this chapter is an example of a boundary value problem.