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Functions of several variables

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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:

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:

See Courant(1950) for details.

Dirichlet's principle

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It is often sufficient to consider only small displacements of the membrane, whose energy difference from no displacement is approximated by

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 must vanish:

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

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

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

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

among all functions φ that satisfy and 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

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

This corresponds to an external force density in D, an external force on the boundary C, and elastic forces with modulus 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 is given by

If we apply the divergence theorem, the result is

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

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

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 vanishes identically on C. In such a case, we could allow a trial function , where c is a constant. For such a trial function,

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

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).