Dirichlet's principle

The Dirichlet principle in potential theory is that functions in an area (with default values ​​on the boundary of ) that the " energy functional " ( Dirichlet integral)

Minimize the Laplace equation

So are harmonic functions in meet.

History

It was Riemannian by Georg Friedrich Bernhard Riemann in support of his theory of surfaces used (in particular for the proof of the existence of analytic functions on these surfaces ), who named it after his teacher Peter Gustav Lejeune Dirichlet. For analytic functions of the real and imaginary parts separately satisfy the Laplace equation. Due to the criticism of Karl Weierstrass, who gave an example of a similar variation issue in which there was no function that took the minimum, the Dirichlet principle was discredited in the 19th century. Only in particular by the work of David Hilbert ( 1904), used the so-called " direct methods " of the calculus of variations, it was rehabilitated and often, for example, by Richard Courant used in the theory of conformal mappings and in the theory of minimal surfaces.

The Dirichlet principle provides a method for the solution of the fundamental for the mathematical physics " Dirichlet problem ", namely the Laplace equation in a given area at given values ​​of the function on the boundary ( Dirichlet boundary condition) to solve. This problem is now characterized namely finding a minimizer of a suitable functional. The latter issue is part of the mathematical field of calculus of variations.

Sketch of proof

Be an arbitrary continuously differentiable function on the boundary of. Then for all

In particular, the limit exists

Since the functional takes in a minimum, is for ever and ever. So, the limit must be 0, ie

The first Green's formula gives

Was being used on the edge.

As to the restrictions mentioned above was arbitrary, it follows from the fundamental lemma of the calculus of variations that the Laplace equation must fulfill.

Caution: Provided been here that we knew a priori that is twice continuously differentiable and that, in the field of the Gaussian integral theorem. The latter is not a big restriction, however, the first implicit assumption is delikaterer nature.