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Functions of several variables

Variational problems that involve multiple integrals arise in numerous applications. For example, if φ(x,y) denotes the displacement of a membrane above the domain D in the x,y plane, then its potential energy is proportional to its surface area:

$U[\varphi ]=\iint _{D}{\sqrt {1+\nabla \varphi \cdot \nabla \varphi }}dx\,dy.\,$

Plateau's problem consists in finding a function that minimizes the surface area while assuming prescribed values on the boundary of D; the solutions are called minimal surfaces. The Euler-Lagrange equation for this problem is nonlinear:

$\varphi _{xx}(1+\varphi _{y}^{2})+\varphi _{yy}(1+\varphi _{x}^{2})-2\varphi _{x}\varphi _{y}\varphi _{xy}=0.\,$

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

$V[\varphi ]={\frac {1}{2}}\iint _{D}\nabla \varphi \cdot \nabla \varphi \,dx\,dy.\,$

The functional V is to be minimized among all trial functions φ that assume prescribed values on the boundary of D. If u is the minimizing function and v is an arbitrary smooth function that vanishes on the boundary of D, then the first variation of $V[u+\epsilon v]$ must vanish:

${\frac {d}{d\epsilon }}V[u+\epsilon v]|_{\epsilon =0}=\iint _{D}\nabla u\cdot \nabla v\,dx\,dy=0.\,$

Provided that u has two derivatives, we may apply the divergence theorem to obtain

$\iint _{D}\nabla \cdot (v\nabla u)\,dx\,dy=\iint _{D}\nabla u\cdot \nabla v+v\nabla \cdot \nabla u\,dx\,dy+\int _{C}v{\frac {\partial u}{\partial n}}ds,\,$

where C is the boundary of D, s is arclength along C and $\partial u/\partial n$ is the normal derivative of u on C. Since v vanishes on C and the first variation vanishes, the result is

$\iint _{D}v\nabla u\cdot \nabla u\,dx\,dy=0\,$

for all smooth functions v that vanish on the boundary of D. The proof for the case of one dimensional integrals may be adapted to this case to show that

$\nabla \cdot \nabla u=0\,$ in D.

The difficulty with this reasoning is the assumption that the minimizing function u 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 Dirichlet's principle in honor of his teacher Dirichlet. However Weierstrass gave an example of a variational problem with no solution: minimize

$W[\varphi ]=\int _{-1}^{1}(x\varphi ')^{2}\,dx\,$

among all functions φ that satisfy $\varphi (-1)=-1$ and $\varphi (1)=1.$ W 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 W=0. The resulting controversy over the validity of Dirichlet's principle is explained in http://www.meta-religion.com/Mathematics/Biography/riemann.htm. Eventually it was shown that Dirichlet's principle is valid, but it requires a sophisticated application of the regularity theory for elliptic partial differential equations; see Jost and Li-Jost (1998).

Generalization to other boundary value problems

A more general expression for the potential energy of a membrane is

$v[\varphi ]=\iint _{D}\left[{\frac {1}{2}}\nabla \varphi \cdot \nabla \varphi +f(x,y)\varphi \right]\,dx\,dy\,+\int _{C}\left[{\frac {1}{2}}\sigma (s)\varphi ^{2}+g(s)\varphi \right]\,ds.$

This corresponds to an external force density $f(x,y)$ in D, an external force $g(s)$ on the boundary C, and elastic forces with modulus $\sigma (s)$ acting on C. The function that minimizes the potential energy with no restriction on its boundary values will be denoted by u. Provided that f and g are continuous, regularity theory implies that the minimizing function u will have two derivatives. In taking the first variation, no boundary condition need be imposed on the increment v. The first variation of $V[u+\epsilon v]$ is given by

$\iint _{D}\left[\nabla u\cdot \nabla v+fv\right]\,dx\,dy+\int _{C}\left[\sigma uv+gv\right]\,ds=0.\,$

If we apply the divergence theorem, the result is

$\iint _{D}\left[-v\nabla \cdot \nabla u+vf\right]\,dx\,dy+\int _{C}v\left[{\frac {\partial u}{\partial n}}+\sigma u+g\right]\,ds=0.\,$

If we first set v=0 on C, the boundary integral vanishes, and we conclude as before that

$-\nabla \cdot \nabla u+f=0\,$

in D. Then if we allow v to assume arbitrary boundary values, this implies that u must satisfy the boundary condition

${\frac {\partial u}{\partial n}}+\sigma u+g=0,\,$

on C. Note that this boundary condition is a consequence of the minimizing property of u: it is not imposed beforehand. Such conditions are called natural boundary conditions.

The preceding reasoning is not valid if $\sigma$ vanishes identically on C. In such a case, we could allow a trial function $\varphi \equiv c$, where c is a constant. For such a trial function,

$V[c]=c\left[\iint _{D}f\,dx\,dy+\int _{C}gds\right].$

By appropriate choice of c, V can assume any value unless the quantity insider the brackets vanishes. Therefore the variational problem is meaningless unless

$\iint _{D}f\,dx\,dy+\int _{C}g\,ds=0.\,$

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).