Dirichlet's Principle

Theorem, in English books attributed to Lord Kelvin, "that there always exists one, but no other than this one, function, v, of x, y, z, which, (1) is finite, continuous, and single-valued, together with its first space derivatives. throughout a given closed region L; (2) at every point of the region satisfies the equation Vov = 0 (see POTENTIAL FUNCTION); and (3) at every point on the boundary of the region has any arbitrarily a igned value, provided this can be regarded as the value at that point of a single-valued function which has derivatives finite, continuous, and single-valued all over this boundary."