Editing Calculus of variations (section)

=== Dirichlet's principle ===
It is often sufficient to consider only small displacements of the membrane, whose energy difference from no displacement is approximated by
<math display="block">V[\varphi] = \frac{1}{2}\iint_D \nabla \varphi \cdot \nabla \varphi \, dx\, dy.</math>
The functional <math>V</math> is to be minimized among all trial functions <math>\varphi</math> that assume prescribed values on the boundary of <math>D.</math> If <math>u</math> is the minimizing function and <math>v</math> is an arbitrary smooth function that vanishes on the boundary of <math>D,</math> then the first variation of <math>V[u + \varepsilon v]</math> must vanish:
<math display="block">\left.\frac{d}{d\varepsilon} V[u + \varepsilon v]\right|_{\varepsilon=0} = \iint_D \nabla u \cdot \nabla v \, dx\,dy = 0.</math>
Provided that u has two derivatives, we may apply the divergence theorem to obtain
<math display="block">\iint_D \nabla \cdot (v \nabla u) \,dx\,dy =
\iint_D \nabla u \cdot \nabla v + v \nabla \cdot \nabla u \,dx\,dy = \int_C v \frac{\partial u}{\partial n} \, ds,</math>
where <math>C</math> is the boundary of <math>D,</math> <math>s</math> is arclength along <math>C</math> and <math>\partial u / \partial n</math> is the normal derivative of <math>u</math> on <math>C.</math> Since <math>v</math> vanishes on <math>C</math> and the first variation vanishes, the result is
<math display="block">\iint_D v\nabla \cdot \nabla u \,dx\,dy =0 </math>
for all smooth functions <math>v</math> that vanish on the boundary of <math>D.</math> The proof for the case of one dimensional integrals may be adapted to this case to show that
<math display="block">\nabla \cdot \nabla u= 0 </math>in <math>D.</math>

The difficulty with this reasoning is the assumption that the minimizing function <math>u</math> must have two derivatives. Riemann argued that the existence of a smooth minimizing function was assured by the connection with the physical problem: membranes do indeed assume configurations with minimal potential energy. Riemann named this idea the [[Dirichlet principle]] in honor of his teacher [[Peter Gustav Lejeune Dirichlet]]. However Weierstrass gave an example of a variational problem with no solution: minimize
<math display="block">W[\varphi] = \int_{-1}^{1} (x\varphi')^2 \, dx</math>
among all functions <math>\varphi</math> that satisfy <math>\varphi(-1)=-1</math> and <math>\varphi(1)=1.</math>
<math>W</math> can be made arbitrarily small by choosing piecewise linear functions that make a transition between −1 and 1 in a small neighborhood of the origin. However, there is no function that makes <math>W=0.</math>{{efn|The resulting controversy over the validity of Dirichlet's principle is explained by Turnbull.<ref>{{cite web |url=http://turnbull.mcs.st-and.ac.uk/~history/Biographies/Riemann.html |title=Riemann biography |publisher=U. St. Andrew |place=UK |author=Turnbull}}</ref>}} Eventually it was shown that Dirichlet's principle is valid, but it requires a sophisticated application of the regularity theory for [[elliptic partial differential equation]]s; see Jost and Li–Jost (1998).