Critical points of harmonic functions on domains in ${\bf R}\sp{3}$
Robert Orrin Shelton · Transactions of the American Mathematical Society · 1980
It is shown that the critical point relations of Morse theory, together with the maximum principle, comprise a complete set of critical point relations for harmonic functions of three variables. The proof proceeds by first constructing a simplified example and then developing techniques to modify this example to realize all admissible possibilities. Techniques used differ substantially from those used by Morse in his solution of the analogous two-variable problem.