Editing Calculus of variations (section)

=== 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).