Content deleted Content added
→Formal statement: Fixed notational inconsistencies in the math and some language. |
→Formal statement: typos and gaps |
||
Line 42:
{{bi|left=1.6|<math>\displaystyle K = \left\{v\in H^1(D): v|_{\partial D} = f\text{ and } v \geq \varphi \right\},</math>}}
which is a [[closed set|closed]] [[convex set|convex]] [[subset]] of the [[Sobolev space]] <math>H^1(D)</math> of square [[integrable function]]s with domain <math>D</math> whose [[weak derivative|weak first derivatives]] is square integrable, containing those functions with the desired boundary conditions and whose values above the obstacle's. A solution to the obstacle problem is a function <math>u
{{bi|left=1.6|<math>\displaystyle J(v) = \int_D |\nabla v|^2\mathrm{d}x</math>}}
over all functions <math>v</math> belonging to <math>K</math>; in symbols
:<math>J(u)=\operatorname{min}_{v\in K} J(v)\text{ or }u\in\operatorname{Argmin}_K J.</math>
The existence and uniqueness of such a minimizer is assured by considerations of [[Hilbert space]] theory.<ref name="Caf383"/><ref>See {{harvnb|Kinderlehrer|Stampacchia|1980|pp=40–41}}.</ref>
==Alternative formulations==
| |||