Shortly after his work on the foundations of geometry David Hilbert turned his attention to finding a suitable statement of the Dirichlet principle, from which to prove the Riemann mapping theorem and vindicate the topological program for complex analysis. Based on comments made in letters to Frege a major motivation for Hilbert's foray into geometry and independence proofs was to investigate the Archimedean axiom. Specifically, Hilbert mentions (to Frege) Dehn's dissertation on the Archimedean axiom and Legendre's theorem. This leads me to think that conformal mappings were on Hilbert's mind and to guess that Weierstrass's counterexample somehow concerned the Archimedean property. But I can't find anything in the secondary history/philosophy of math literature that quite puts all the pieces of the puzzle together--that Weierstrass had a counter-example is mentioned but details are skirted--and qua philosopher I'm bumping up against my mathematical horizons in piecing it together myself.
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Weierstrass simply observed that not every problem in the calculus of variations would have a solution. He considered the example $$D[y]=\int_{-1}^{1}x^2\left(\frac{d y}{dx}\right)^2dx\to \min,$$ where the functional $D[y]$ is minimized over continuous functions having piecewise continuous first derivatives in $[-1,1]$ and satisfying the boundary conditions $y(-1)=0$, $y(1)=1$. He proved that although there is a minimizing sequence $y_n=y_n(.)$ which makes $D[y_n]$ arbitrarily small, the minimal value of zero is never actually attained. Weierstrass's example called into question the a priori validity of Dirichlet's principle. However, it did not completely refute the specific applications of Dirichlet's principle to boundary value problems for Laplace's equation developed by Green, Dirichlet, Riemann and others. It simply implied that the particular result required by Riemann would need a formal proof, which Riemann had not provided. For that reason some people refer to this example as Weierstrass's critique rather than Weierstrass's counterexample. The story is briefly discussed in "A History of Analysis" edited by Hans Niels Jahnke. |
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I think I vaguely remember what the counterexpample was, but not the details. So if someone can fill it in it'd be great! (I'm putting this in CW mode for that reason.) The idea is based on knowing explicitly the Green's function in the disk. The goal is to construct a continuous function $g$ on the boundary of the unit disk, such that it is the trace of some smooth function $f$ where $\triangle f = 0$ in the disk, and where the energy integral $\int_D |\partial f|^2 dx = \infty$. The construction itself, I think, was based on finding a sequence of harmonic functions $f_k$ such that in the interior of the disk $\sum \partial^\alpha f_k$ converges pointwise for any derivative with multiindex $\alpha$ (but of course the sum fails to converge for any derivative on the boundary, so the final function only extends continuously to the boundary). the Trick is that these functions are chosen with specific boundary values, so that using the Green's function one can estimate (from below) the size of their gradients near the boundary. Then one just need to make sure that the blow-up rate of the gradient dominates the distance to the boundary, so the energy integral will fail to converge in a neighborhood of the boundary. Since the energy integral of the solution to the Euler-Lagrange equation is infinite, the solution obviously is not a minimizer of the energy integral, thus contradicting Dirichlet's principle. |
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A short introduction and connection to Jacob Steiner's proofs of Dido's Problem can be found in Perron, O., Zur Existenzfrage eines Maximums oder Minimums, Deutsche Math.-Ver. 22, 140-144 (1913). |
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