Editing Calculus of variations (section)

== Functions of several variables ==

For example, if <math>\varphi(x, y)</math> denotes the displacement of a membrane above the domain <math>D</math> in the <math>x,y</math> plane, then its potential energy is proportional to its surface area:
<math display="block">U[\varphi] = \iint_D \sqrt{1 +\nabla \varphi \cdot \nabla \varphi} \,dx\,dy.</math>
[[Plateau's problem]] consists of finding a function that minimizes the surface area while assuming prescribed values on the boundary of <math>D</math>; the solutions are called '''minimal surfaces'''. The Euler–Lagrange equation for this problem is nonlinear:
<math display="block">\varphi_{xx}(1 + \varphi_y^2) + \varphi_{yy}(1 + \varphi_x^2) - 2\varphi_x \varphi_y \varphi_{xy} = 0.</math>
See Courant (1950) for details.

=== 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).

=== Generalization to other boundary value problems ===
A more general expression for the potential energy of a membrane is
<math display="block">V[\varphi] = \iint_D \left[ \frac{1}{2} \nabla \varphi \cdot \nabla \varphi + f(x,y) \varphi \right] \, dx\,dy \, + \int_C \left[ \frac{1}{2} \sigma(s) \varphi^2 + g(s) \varphi \right] \, ds.</math>
This corresponds to an external force density <math>f(x,y)</math> in <math>D,</math> an external force <math>g(s)</math> on the boundary <math>C,</math> and elastic forces with modulus <math>\sigma(s)</math>acting on <math>C.</math> The function that minimizes the potential energy '''with no restriction on its boundary values''' will be denoted by <math>u.</math> Provided that <math>f</math> and <math>g</math> are continuous, regularity theory implies that the minimizing function <math>u</math> will have two derivatives. In taking the first variation, no boundary condition need be imposed on the increment <math>v.</math> The first variation of <math>V[u + \varepsilon v]</math> is given by
<math display="block">\iint_D \left[ \nabla u \cdot \nabla v + f v \right] \, dx\, dy + \int_C \left[ \sigma u v + g v \right] \, ds = 0. </math>
If we apply the divergence theorem, the result is
<math display="block">\iint_D \left[ -v \nabla \cdot \nabla u + v f \right] \, dx \, dy + \int_C v \left[ \frac{\partial u}{\partial n} + \sigma u + g \right] \, ds =0. </math>
If we first set <math>v = 0</math> on <math>C,</math> the boundary integral vanishes, and we conclude as before that
<math display="block">- \nabla \cdot \nabla u + f =0 </math>
in <math>D.</math> Then if we allow <math>v</math> to assume arbitrary boundary values, this implies that <math>u</math> must satisfy the boundary condition
<math display="block">\frac{\partial u}{\partial n} + \sigma u + g =0, </math>
on <math>C.</math> This boundary condition is a consequence of the minimizing property of <math>u</math>: it is not imposed beforehand. Such conditions are called '''natural boundary conditions'''.

The preceding reasoning is not valid if <math>\sigma</math> vanishes identically on <math>C.</math> In such a case, we could allow a trial function <math>\varphi \equiv c,</math> where <math>c</math> is a constant. For such a trial function,
<math display="block">V[c] = c\left[ \iint_D f \, dx\,dy + \int_C g \, ds \right].</math>
By appropriate choice of <math>c,</math> <math>V</math> can assume any value unless the quantity inside the brackets vanishes. Therefore, the variational problem is meaningless unless
<math display="block">\iint_D f \, dx\,dy + \int_C g \, ds =0.</math>
This condition implies that net external forces on the system are in equilibrium. If these forces are in equilibrium, then the variational problem has a solution, but it is not unique, since an arbitrary constant may be added. Further details and examples are in Courant and Hilbert (1953).