The criticism of Riemann's argument

The criticism of Riemann's argument was in two directions. First, for functionals apparently similar to the Dirichlet integral it was shown that no minimizer exists. On the other hand, F. Prym, in 1871, gave an example of a continuous boundary datum defined on the circle for which no extension in the disc has finite energy. Thus, the legitimacy of Riemann's Dirichlet principle as a tool for proving existence of harmonic functions was put in serious doubt for several decades. This program was reinstated as a major theme of mathematical research by Hilbert in 1900 and gave rise to an extensive development of methods in this domain (see Section 6).