The Dirichlet problem can be solved for many PDEs, although originally it was posed for Laplace's equation. In that case the problem can be stated as follows:
Given a function f that has values everywhere on the boundary of a region in Rn, is there a unique continuous functionu twice continuously differentiable in the interior and continuous on the boundary, such that u is harmonic in the interior and u = f on the boundary?
The Dirichlet problem goes back to George Green, who studied the problem on general domains with general boundary conditions in his Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism, published in 1828. He reduced the problem into a problem of constructing what we now call Green's functions, and argued that Green's function exists for any domain. His methods were not rigorous by today's standards, but the ideas were highly influential in the subsequent developments. The next steps in the study of the Dirichlet's problem were taken by Karl Friedrich Gauss, William Thomson (Lord Kelvin) and Peter Gustav Lejeune Dirichlet, after whom the problem was named, and the solution to the problem (at least for the ball) using the Poisson kernel was known to Dirichlet (judging by his 1850 paper submitted to the Prussian academy). Lord Kelvin and Dirichlet suggested a solution to the problem by a variational method based on the minimization of "Dirichlet's energy". According to Hans Freudenthal (in the Dictionary of Scientific Biography, vol. 11), Bernhard Riemann was the first mathematician who solved this variational problem based on a method which he called Dirichlet's principle. The existence of a unique solution is very plausible by the "physical argument": any charge distribution on the boundary should, by the laws of electrostatics, determine an electrical potential as solution. However, Karl Weierstrass found a flaw in Riemann's argument, and a rigorous proof of existence was found only in 1900 by David Hilbert, using his direct method in the calculus of variations. It turns out that the existence of a solution depends delicately on the smoothness of the boundary and the prescribed data.
For a domain having a sufficiently smooth boundary , the general solution to the Dirichlet problem is given by
is the derivative of the Green's function along the inward-pointing unit normal vector . The integration is performed on the boundary, with measure. The function is given by the unique solution to the Fredholm integral equation of the second kind,
The Green's function to be used in the above integral is one which vanishes on the boundary:
for and . Such a Green's function is usually a sum of the free-field Green's function and a harmonic solution to the differential equation.
The Dirichlet problem for harmonic functions always has a solution, and that solution is unique, when the boundary is sufficiently smooth and is continuous. More precisely, it has a solution when
Example: equation of a finite string attached to one moving wall
Consider the Dirichlet problem for the wave equation describing a string attached between walls with one end attached permanently and the other moving with the constant velocity i.e. the d'Alembert equation on the triangular region of the Cartesian product of the space and the time:
As one can easily check by substitution, the solution fulfilling the first condition is
Bers, Lipman; John, Fritz; Schechter, Martin (1979), Partial differential equations, with supplements by Lars Gȧrding and A. N. Milgram, Lectures in Applied Mathematics, vol. 3A, American Mathematical Society, ISBN0-8218-0049-3.