Or did Riemann take for granted the existence of a function that achieves the minimum value under the special assumption that used the Dirichlet principle (the assumption of partial smoothness as stated in Riemann's doctoral thesis)?


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Riemann did not quite take it for granted and gave an argument that, in his opinion, showed existence of the minimizing harmonic function, but, in fact, it did not. Klein in Development of Mathematics in the 19th Century gives a heuristic physical construction (electric potential on a surface with charges placed at isolated points) that possibly motivated Riemann. The story with Weierstrass's famous criticism is somewhat oversimplified, as Weierstrass's counterexample was not of an energy integral considered by Riemann. But his general point that the assertion of existence in such cases, which Dirichlet deemed "clear", could be false was valid. The details of Riemann's use are explored in Gray, On the history of the Riemann mapping theorem:

"It is often said that Riemann's proof rested at a crucial point on an appeal to Dirichlet's principle, and that for this reason it was not widely accepted... Hilbert claimed that considerations of this nature had led Riemann to his proof of the existence of functions with given boundary values, but that Weierstrass was the first to show that this approach was not reliable... In fact Riemann's approach was rather different and, more to the point, was soon shown to be fatally flawed.

Riemann did not naively apply something called Dirichlet's principle, if this principle is taken to be the claim that a continuous function defined on the boundary of a simply connected region extends to a harmonic function defined on the whole region. Rather, Riemann first asserted that if a certain integral over a surface $T$... is finite the integral attains a minimal value and moreover this minimum is attained by a unique function if one excludes the points of discontinuity. This unique minimizing function is harmonic. The condition on the integral is certainly not naive, even if, as we shall see, it is inadequate to ensure the purpose.

Riemann then considered a function $\lambda$ that vanished on the boundary, could be discontinuous at isolated points, and for which the integral (later called the Dirichlet integral by Hilbert) $L(\lambda)$... is finite. "The totality of these functions $\lambda$," he wrote, "represents a connected domain closed in itself, in which each function can be transformed continuously into every other, and a function cannot approach indefinitely closely to one which is discontinuous along a curve without $L(\lambda)$ becoming infinite"... So Riemann's use of the Dirichlet's principle rests on a claim that certain boundary behaviour is sufficient to guarantee that a certain integral, $L$, is always finite, and that therefore the [related] integral $\Omega$ is also finite and attains its minimum."


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