Dirichlet principle

The Dirichlet principle in potential theory says that functions exist in an area (with given continuous values on the edge of ) that define the "energy functional" (Dirichlet integral) ${\displaystyle u}$ ${\displaystyle G\subset \mathbb {R} ^{n}}$ ${\displaystyle u=g}$ ${\displaystyle G}$

${\displaystyle D(u)=\int _{G}\left({\left({\frac {\partial u}{\partial x_{1}}}\right)}^{2}+{\left({\frac {\partial u}{\partial x_{2}}}\right)}^{2}+\dots +{\left({\frac {\partial u}{\partial x_{n}}}\right)}^{2}\right)\,\mathrm {d} \lambda ^{n}=\int _{G}|\nabla u|^{2}\mathrm {d} \lambda ^{n}\geq 0}$

minimize, and the Laplace equation

${\displaystyle \Delta u=0}$

in meet, so harmonic functions are. It is assumed that the functions in and on the boundary of are continuous and are continuously differentiable in ( , see for and differentiation class ). Sometimes a statement of uniqueness for the function (and the minimum of the Dirichlet integral) is also added. ${\displaystyle G}$ ${\displaystyle u}$ ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle u\in C^{0}({\bar {G}})\cap C^{1}(G)}$ ${\displaystyle C^{0}}$ ${\displaystyle C^{1}}$

history

It was used by Georg Friedrich Bernhard Riemann to justify his theory of Riemann surfaces (especially to prove the existence of analytical functions on these surfaces), who named it after his teacher Peter Gustav Lejeune Dirichlet . Although it does not appear explicitly in Dirichlet's writings, it was used by him in his lectures, from which Riemann knew it. In the case of analytical functions, the real and imaginary parts separately satisfy the Laplace equation. The Dirichlet principle was discredited in the 19th century by the criticism of Karl Weierstrass , who gave an example of a similar problem of variation in which there was no function that assumed the minimum. Only through the work of David Hilbert (1904), who used so-called “direct methods” of the calculus of variations, was it rehabilitated and then often e.g. B. used by Richard Courant in the theory of conformal mappings and in the theory of minimal surfaces.

The Dirichlet principle provides a method for solving the “ Dirichlet problem ”, which is fundamental for mathematical physics , namely to solve the Laplace equation in a given area for given values ​​of the function on the boundary ( Dirichlet boundary condition ). This problem is namely now characterized by finding a minimizer for a suitable functional. The latter question belongs to the mathematical field of the calculus of variations . ${\displaystyle G}$

Dirichlet was aware of the conception of the Dirichlet integral as potential energy and that the functions that minimize the Dirichlet integral correspond to the equilibrium positions of a system. Since the Dirichlet integral is greater than or equal to zero, the existence of a minimal solution was considered evident. Dirichlet's principle was also preceded by William Thomson and Carl Friedrich Gauß .

Evidence sketch

Let be any continuously differentiable function with on the boundary of . Then applies to everyone ${\displaystyle v}$ ${\displaystyle v=0}$ ${\displaystyle G}$ ${\displaystyle \varepsilon \in \mathbb {R} }$

${\displaystyle D(u+\varepsilon v)=D(u)+2\varepsilon \int _{G}(\nabla u)(\nabla v)\mathrm {d} \lambda ^{n}+{\varepsilon }^{2}D(v)\ .}$

In particular, the Limes exists

${\displaystyle \lim _{\varepsilon \rightarrow 0}{\frac {D(u+\varepsilon v)-D(u)}{\varepsilon }}=2\int _{G}(\nabla u)(\nabla v)\mathrm {d} \lambda ^{n}\ .}$

Since the functional in is at a minimum, is for and for . So the limit must be 0, ie ${\displaystyle D}$ ${\displaystyle u}$ ${\displaystyle {\frac {D(u+\varepsilon v)-D(u)}{\varepsilon }}\geq 0}$ ${\displaystyle \varepsilon >0}$ ${\displaystyle {\frac {D(u+\varepsilon v)-D(u)}{\varepsilon }}\leq 0}$ ${\displaystyle \varepsilon <0}$

${\displaystyle 0=2\int _{G}(\nabla u)(\nabla v)\mathrm {d} \lambda ^{n}\ .}$

The first Green formula delivers

${\displaystyle 0=\int _{G}(\nabla u)(\nabla v)\mathrm {d} \lambda ^{n}=-\int _{G}(v\triangle u)\mathrm {d} \lambda ^{n}+\int _{\partial G}v{\frac {\partial u}{\partial \nu }}\,\mathrm {d} \sigma =-\int _{G}(v\triangle u)\mathrm {d} \lambda ^{n}\ ,}$

where was used on the edge . ${\displaystyle v=0}$ ${\displaystyle \partial G}$

Since there was arbitrary except for the above restrictions, it follows from the fundamental lemma of the calculus of variations that the Laplace equation must satisfy in. ${\displaystyle v}$ ${\displaystyle u}$ ${\displaystyle G}$

${\displaystyle \Box }$

Caution : It was assumed here that one knew a priori that twice is continuously differentiable and that the Gaussian integral theorem applies in this area. The latter is not a major restriction, but the first implicit requirement is more delicate. ${\displaystyle u}$ ${\displaystyle G}$

literature

• Lars Gårding : The Dirichlet problem. In: Mathematical Intelligencer. 2, No. 1, 1979, ISSN  0343-6993 , pp. 42-52.
• Stefan Hildebrandt : Comments on Dirichlet's principle. In: Hermann Weyl : The idea of ​​the Riemann surface. Teubner, Leipzig et al. 1913, p. 197 ( Mathematical lectures at the University of Göttingen 5, ZDB -ID 978485-8 ), (Reprint expanded with an appendix. Edited by Reinhold Remmert . Teubner, Stuttgart et al. 1997, ISBN 3-8154-2096 -2 ( Teubner Archive for Mathematics. Supplement 5)).
• AF Monna : Dirichlet's Principle. A mathematical comedy of errors and its influence on the development of analysis. Oosthoek, Scheltema & Holkema, Utrecht 1975, ISBN 90-313-0175-2 .
• Richard Courant : Dirichlet's Principle, Conformal Mapping and Minimal Surfaces , Interscience 1950

References and comments

1. ↑ This also includes the statement that in twice is continuously differentiable ${\displaystyle u}$ ${\displaystyle G}$
2. ↑ Courant, Dirichlet's principle, Conformal Mapping and Minimal Surfaces, 1950, p. 6
3. ^ Dirichlet's Principle , Mathworld
4. ↑ Hildebrandt, Remarks on Dirichlet's Principle, in: Weyl, The Idea of ​​the Riemann Surface, Springer 1997, p. 197
5. ↑ Hildebrandt, loc. cit., p. 197
6. ↑ Hildebrandt, loc. cit., p. 199
7. ^ Courant, Dirichlet's principle, conformal mappings and minimal surfaces, 1905, p. 2