Dirichlet Problem

The problem of finding the connection between a continuous function f on the boundary partialR of a region R with a harmonic function taking on the value f on partialR. 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 f is a continuous function on the boundary of the open unit disk partialD(0,1), then define

 u(z)={1/(2pi)int_0^(2pi)f(e^(ipsi))(1-|z|^2)/(|z-e^(ipsi)|^2)dpsi   if z in D(0,1); f(z)   if z in partialD(0,1),

where partialD(0,1) is the boundary of D(0,1). Then u is continuous on the closed unit disk D^_(0,1) and harmonic on D(0,1) (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).

See also

Poisson Integral, Poisson Kernel

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Courant, R. and Hilbert, D. Methods of Mathematical Physics, Vol. 1. New York: Wiley, pp. 179-180 and 240, 1989.Ebenfelt, P. and Viscardi, M. "On the Solution of the Dirichlet Problem with Rational Holomorphic Boundary Data." Comput. Meth. Func. Th. 5, 445-457, 2005., S. G. "The Dirichlet Problem" and "Application of Conformal Mapping to the Dirichlet Problem." §7.3.3, 7.7.1, and 14.2 in Handbook of Complex Variables. Boston, MA: Birkhäuser, pp. 93, 97-98, and 164-168, 1999a.Krantz, S. G. A Panorama of Harmonic Analysis. Washington, DC: Math. Assoc. Amer., 1999b.Logan, J. D. Applied Mathematics, 2nd ed. New York: Wiley, 1997.Mathematical Association of America. "Mathematics Student Wins the Siemens-Westinghouse Competition." Jan. 9, 2006. Foundation. "2005 Siemens Competition Nationals."

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Dirichlet Problem

Cite this as:

Weisstein, Eric W. "Dirichlet Problem." From MathWorld--A Wolfram Web Resource.

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