The problem of finding the connection between a continuous function f on the boundary dR of a region R with a harmonic function taking on the value f on dR. In general, the problem asks if such a solution exists and, if so, if it is unique. The Dirichlet problem is extremely important in mathematical physics (Courant and Hilbert 1989, pp. 179-180 and 240; Logan 1997; Krantz 1999b).

Poisson integral | Poisson kernel