I came accros this two exercises on existence of minimizers:
(Pointwise gradient constraint) It is exercise 15 in chapter 8 in Evans PDE book
(a) Show there exists a unique minimizer $u \in A$ of
$I[w]=\int_{\Omega}(\frac{1}{2}|\nabla u|^2-fw) dx$
where $f \in L^2(\Omega)$ and $A=\{u \in W^{1,2}_0(\Omega), |\nabla u|\leq 1$ a.e. in $\Omega\}$.
(b) Prove $\int_\Omega \nabla u \nabla(w-u)dx\geq \int_\Omega f(w-u)dx$ for any $w\in A$
This is second exercise:
Let $U_0 \in W^{1,2}(\Omega)$ and let $u_0$ be the trace of $U_0$ on $\partial \Omega$, where $\Omega \in C^{0,1}$ is an open subset of $\mathbb{R}^d$. Let $A=\{u \in W^{1,2}(\Omega), u=u_0$ on $\partial \Omega \}$. Let $a_{ij} \in L^\infty (\Omega),i,j=1,2,\dots,d $ ,fulfill the ellipticity condition. Let $f \in L^2(\Omega)$. Show that there exists a unique minimizer of
$I[w]=\int_{\Omega}(\sum_{i,j=1}^d a_{ij} \frac{\partial w}{\partial x_j}\frac{\partial w}{\partial x_i}-fw) dx$
in $A$. Deduce the corresponding Euler Lagrange equation and show that the minimizer is the unique solution to this equation.