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General solution

For a domain D {\displaystyle D} having a sufficiently smooth boundary ∂ D {\displaystyle \partial D}, the general solution to the Dirichlet problem is given by

where G (x, y) {\displaystyle G(x,y)} is the Green's function for the partial differential equation, and

is the derivative of the Green's function along the inward-pointing unit normal vector n ^ {\displaystyle {\widehat {n}}}. The integration is performed on the boundary, with measure d s {\displaystyle ds}. The function ν (s) {\displaystyle \nu (s)} 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 s ∈ ∂ D {\displaystyle s\in \partial D} and x ∈ D {\displaystyle x\in D}. Such a Green's function is usually a sum of the free-field Green's function and a harmonic solution to the differential equation.