Talk:Dirichlet principle

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Geometry guy 15:59, 10 June 2007 (UTC)

[edit] What are the constraints on the boundary?

It sounds like the principle holds in only certain cases, but its not clear what these are. What can be said about fractal boundaries? (i.e. no-where differentiable boundaries)? linas 03:23, 26 August 2006 (UTC)

I think you need to be able to integrate by parts and you should have good trace and extension operators. I am pretty certain that a Lipschitz boundary for $\partial\Omega$ and $g\in H^{1/2}(\partial\Omega)$ are sufficient. Kusma (討論) 06:09, 26 August 2006 (UTC)

[edit] What example?

The article says "Weierstraß gave an example of a functional that does not attain its minimum. Hilbert later justified Riemann's use of Dirichlet's principle." What's the example? —Ben FrantzDale 18:08, 19 December 2006 (UTC)

According to Giaquinta/Hildebrandt, Calculus of Variations, vol. I, p. 43, the example Weierstraß gave is

$F(u)=\int_{-1}^1 x^2 (u'(x))^2 dx$

in the set $\mathcal{C}=\{u\in C^1([-1,1]) : u(-1)=-1 \text{ and }u(1)=1 \}$ . It is clear that $\inf_{\mathcal{C}} F\ge 0$ . A minimizing sequence is given by

$u_\varepsilon(x)=\frac{\arctan\tfrac{x}{\varepsilon}}{\arctan\tfrac{1}{\varepsilon}}\in \mathcal{C}$ .

For $\varepsilon\to 0$ , $F(u_\varepsilon)\to 0$ . However, there is no function v such that F(v)=0, since that would imply (as we are asking for $C 1$ smoothness) that the derivative is zero everywhere, a contradiction to the boundary conditions. Note that $u_\varepsilon$ converge to the sign function as $\varepsilon\to 0$ . Hope that helps, Kusma (討論) 18:23, 19 December 2006 (UTC)