The problem of finding the connection between a continuous function on the boundary of a region with a harmonic function taking on the value on . 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)
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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).