An obvious way to obtain the variational formulation of the obstacle problem is to consider the total potential energy:
@f{equation*}
- E(u):=\dfrac{1}{2}\int\limits_{\Omega} \nabla u \cdot \nabla - \int\limits_{\Omega} fu.
+ E(u):=\dfrac{1}{2}\int\limits_{\Omega} \nabla u \cdot \nabla u - \int\limits_{\Omega} fu.
@f}
We have to find a solution $u\in G$ of the following minimization problem:
@f{equation*}
$\mu-\lambda<0$ at points where $\lambda=0$, given the definition of $K$.
The existence and uniqueness of $(u,\lambda)\in V\times K$ of this saddle
-point problem has been stated in Grossmann and Roos: Numerical treatment of
-partial differential equations, Springer-Verlag, Heidelberg-Berlin, 2007, 596
-pages, ISBN 978-3-540-71582-5.
+point problem has been stated in Glowinski, Lions and Tr\'{e}moli\`{e}res: Numerical Analysis of Variational
+Inequalities, North-Holland, 1981.