Editing Dirichlet problem (section)

==General solution==
For a domain <math>D</math> having a sufficiently smooth boundary <math>\partial D</math>, the general solution to the Dirichlet problem is given by

: <math>u(x) = \int_{\partial D} \nu(s) \frac{\partial G(x, s)}{\partial n} \,ds,</math>

where <math>G(x, y)</math> is the [[Green's function]] for the partial differential equation, and

: <math>\frac{\partial G(x, s)}{\partial n} = \widehat{n} \cdot \nabla_s G (x, s) = \sum_i n_i \frac{\partial G(x, s)}{\partial s_i}</math>

is the derivative of the Green's function along the inward-pointing unit normal vector <math>\widehat{n}</math>. The integration is performed on the boundary, with [[Measure (mathematics)|measure]] <math>ds</math>. The function <math>\nu(s)</math> is given by the unique solution to the [[Fredholm integral equation]] of the second kind,

: <math>f(x) = -\frac{\nu(x)}{2} + \int_{\partial D} \nu(s) \frac{\partial G(x, s)}{\partial n} \,ds.</math>

The Green's function to be used in the above integral is one which vanishes on the boundary:

: <math>G(x, s) = 0</math>

for <math>s \in \partial D</math> and <math>x \in D</math>. Such a Green's function is usually a sum of the free-field Green's function and a harmonic solution to the differential equation.

===Existence===
The Dirichlet problem for harmonic functions always has a solution, and that solution is unique, when the boundary is sufficiently smooth and <math>f(s)</math> is continuous. More precisely, it has a solution when

: <math>\partial D \in C^{1,\alpha}</math>

for some <math>\alpha \in (0, 1)</math>, where <math>C^{1,\alpha}</math> denotes the [[Hölder condition]].