The Dirichlet problem asks for a harmonic function on a region 
whose boundary values on 
equal a prescribed continuous
function 
.
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).
If 
is a continuous function on the boundary of
the open unit disk 
,
then define
 |
(1)
|
where 
is the boundary of 
.
Then 
is continuous on the closed unit disk 
and harmonic on 
(Krantz 1999a, p. 93).
For the case of rational boundary data without poles, the resulting solution of the Dirichlet problem is also rational (Ebenfelt and Viscardi 2005), the proof of which led to Viscardi winning the 2005-2006 Siemens-Westinghouse competition (Siemens Foundation 2005; Mathematical Association of America 2006).
