Harnack's first convergence theorem

Harnack's first convergence theorem [′här·naks ¦fərst kən′vər·jəns ‚thir·əm]

(mathematics)

The theorem that if a sequence of functions harmonic in a common domain of three-dimensional space and continuous on the boundary of the domain converges uniformly on the boundary, then it converges uniformly in the domain to a function which is itself harmonic; the sequence of any partial derivative of the functions in the original sequence converges uniformly to the corresponding partial derivative of the limit function in every closed subregion of the domain.