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Since it is the integral of a non-negative quantity, the
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that satisfies the boundary conditions and has minimal
Dirichlet energy.
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186:{\displaystyle E={\frac {1}{2}}\int _{\Omega }\|\nabla u(x)\|^{2}\,dx,}
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just shows that the
Lagrange equations (or, equivalently, the
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Pages displaying wikidata descriptions as a fallback
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Pages displaying wikidata descriptions as a fallback
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Pages displaying wikidata descriptions as a fallback
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Pages displaying wikidata descriptions as a fallback
51:. The Dirichlet energy is intimately connected to
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16:A mathematical measure of a function's variability
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332:and such solutions are the topic of study in
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55:and is named after the German mathematician
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480:. American Mathematical Society.
339:In a more general setting, where
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510:Partial differential equations
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278:{\displaystyle -\Delta u(x)=0}
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319:of finding a function
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224:Properties and applications
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476:Lawrence C. Evans (1998).
410:Hamilton–Jacobi equations
311:, subject to appropriate
442:Bounded mean oscillation
235:for every function
384:. The solutions to the
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422:Dirichlet's principle
392:are those functions
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91:of the function
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388:for the sigma model
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99:real number
21:mathematics
499:Categories
470:References
390:Lagrangian
63:Definition
40:functional
299:Ω
296:∈
255:Δ
252:−
165:‖
149:∇
146:‖
141:Ω
137:∫
67:Given an
37:quadratic
416:See also
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285:for all
209:gradient
69:open set
33:function
29:variable
97:is the
42:on the
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355:, and
196:where
23:, the
482:ISBN
341:Ω ⊆
87:the
72:Ω ⊆
374:→ Φ
233:≥ 0
19:In
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111:E
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84:R
80:u
74:R
48:H
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