User:Donludwig/Sandbox

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
 * $$ \iint_D \left[ \nabla u \cdot \nabla v + f v \right] \, dx\, dy + \int_C \left[ \sigma u v + g v \right] \, ds =0. \,$$

If we apply the divergence theorem, the result is
 * $$ \iint_D \left[ -v \nabla \cdot \nabla u + v f \right] \, dx \, dy + \int_C v \left[ \frac{\partial u}{\partial n} + \sigma u + g \right] \, ds =0. \,$$

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,
 * $$ V[c] = c\left[ \iint_D f \, dx\,dy + \int_C g ds \right].$$

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).