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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
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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
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The article says "WeierstraĂ 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?
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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.
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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)?
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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.
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635:{\displaystyle u_{\varepsilon }(x)={\frac {\arctan {\tfrac {x}{\varepsilon }}}{\arctan {\tfrac {1}{\varepsilon }}}}\in {\mathcal {C}}}
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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
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It is standard notation, but the article might be made more accessible by simply specifying the boundary conditions in words.
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It's standard notation for the boundary of the domain. I believe it's still taught as part of multivariate calculus courses.
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smoothness) that the derivative is zero everywhere, a contradiction to the boundary conditions. Note that
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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is constant, so the limiting function does not satisfy the boundary condition at the origin.
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at all. It is obvious that the integral is nonnegative for harmonic functions (
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501:{\displaystyle {\mathcal {C}}=\{u\in C^{1}():u(-1)=-1{\text{ and }}u(1)=1\}}
996:{\displaystyle E=E+{\frac {1}{2}}\int _{\Omega }|\nabla (u-v)|^{2}dV.}
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An example of a boundary value problem with no solution is to take
1139:{\displaystyle E=-{\frac {1}{2}}\int _{\Omega }|\nabla u|^{2}dV.}
836:) but I do not believe it is nonnegative in the general case. --
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375:{\displaystyle F(u)=\int _{-1}^{1}x^{2}(u'(x))^{2}dx}
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283:, vol. I, p. 43, the example WeierstraĂ gave is
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539:{\displaystyle \inf _{\mathcal {C}}F\geq 0}
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1005:So the functional is bounded below in the
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1566:. Princeton University Press. pp. 49â51.
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1341:{\displaystyle (x^{2}+y^{2})^{\alpha }}
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1031:case also. When the boundary value
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1348:satisfy the boundary conditions and
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546:. A minimizing sequence is given by
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795:{\displaystyle \varepsilon \to 0}
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665:{\displaystyle \varepsilon \to 0}
115:Knowledge:WikiProject Mathematics
1593:Start-Class mathematics articles
1482:{\displaystyle \partial \Omega }
769:{\displaystyle u_{\varepsilon }}
714:. However, there is no function
192:{\displaystyle \partial \Omega }
118:Template:WikiProject Mathematics
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109:and see a list of open tasks.
1393:{\displaystyle \alpha \to 0}
1057:, we can even compute that
870:is no-zero, and a solution
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1184:{\displaystyle \nabla u}
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1210:{\displaystyle \Omega }
1024:{\displaystyle f\neq 0}
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1419:{\displaystyle E=0}
1288:{\displaystyle u=0}
1262:{\displaystyle u=1}
1236:{\displaystyle f=0}
1050:{\displaystyle g=0}
802:. Hope that helps,
508:. It is clear that
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137:Mid-priority
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62:Midâpriority
40:WikiProjects
1534:specified.
385:in the set
112:Mathematics
103:mathematics
59:Mathematics
30:Start-class
1587:Categories
1572:0691079889
1530:potential
1450:Mike Stone
718:such that
1536:catslash
1400:. But
850:When
139:on the
1463:Jargon
838:Yecril
720:F(v)=0
602:arctan
582:arctan
36:scale.
804:Kusma
254:Kusma
164:linas
1568:ISBN
1540:talk
1496:talk
1454:talk
842:talk
807:(èšè«)
646:For
474:and
257:(èšè«)
199:and
176:for
1509:Ray
1374:as
834:⥠0
517:inf
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