Talk:Dirichlet's principle

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)Reply

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 ${\displaystyle \partial \Omega }$  and ${\displaystyle g\in H^{1/2}(\partial \Omega )}$  are sufficient. Kusma (討論) 06:09, 26 August 2006 (UTC)Reply

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)Reply

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

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

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

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

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

The entry for Dirichlet’s energy does not mention the free component f at all. It is obvious that the integral is nonnegative for harmonic functions (f ≡ 0) but I do not believe it is nonnegative in the general case. --Yecril (talk) 10:36, 18 September 2008 (UTC)Reply

When ${\displaystyle f}$  is no-zero, and a solution ${\displaystyle u}$  exists, then an integration by parts shows that

${\displaystyle E[v]=E[u]+{\frac {1}{2}}\int _{\Omega }|\nabla (u-v)|^{2}dV.}$ 

So the functional is bounded below in the ${\displaystyle f\neq 0}$  case also. When the boundary value ${\displaystyle g=0}$ , we can even compute that

${\displaystyle E[u]=-{\frac {1}{2}}\int _{\Omega }|\nabla u|^{2}dV.}$ 

The lower bound is therefore negative whenever ${\displaystyle f}$  is such that ${\displaystyle \nabla u}$  is not identically zero.

It is not immediately obvious to me that a similar lower bound can be written down in the case that there is no solution to the boundary-value problem.

An example of a boundary value problem with no solution is to take ${\displaystyle \Omega }$  to be the punctured unit disc, take ${\displaystyle f=0}$ , and impose boundary conditions that ${\displaystyle u=1}$  on the outer boundary and ${\displaystyle u=0}$  at the origin. The functions ${\displaystyle (x^{2}+y^{2})^{\alpha }}$  satisfy the boundary conditions and ${\displaystyle E\to 0}$  as ${\displaystyle \alpha \to 0}$ . But ${\displaystyle E=0}$  implies that ${\displaystyle u}$  is constant, so the limiting function does not satisfy the boundary condition at the origin.

Mike Stone (talk) 18:50, 8 January 2013 (UTC)Reply

${\displaystyle \partial \Omega }$  means what? Any GOOD reason that so much here is left undefined/unexplained? Are we supposed to know (and if so, based on what) what these symbols mean in this case? I've taken math through ordinary differential equations, so my knowledge is "above average" for a layperson. Most of these symbols have various meanings in various mathematical contexts, and need to be defined prior to indiscriminate use. Writing articles for the specialist seems unproductive in this forum. 71.31.146.16 (talk) 12:26, 19 June 2012 (UTC)Reply

It's standard notation for the boundary of the domain. I believe it's still taught as part of multivariate calculus courses. Ray^Talk 06:37, 20 June 2012 (UTC)Reply

It is standard notation, but the article might be made more accessible by simply specifying the boundary conditions in words.

In any case, it is not true that potential must be specified over the entire boundary of the domain. Dirichlet's principle still holds if the domain has "side walls" on which the potential is free to vary (but has no gradient normal to the wall), provided only that there is somewhere where the potential is specified.^[1] catslash (talk) 01:26, 19 December 2019 (UTC)Reply