Uniqueness of minimizers in a problem in the Calculus of Variations - Part II

Existence of minimizers should not be a severe issue...

The proof of the existence follows form the Arzela-Ascoli theorem, but the proof is not entirely obvious. This is Proposition 1.1 in:

E. Giusti, Direct methods in the calculus of variations. World Scientific Publishing Co., Inc., River Edge, NJ, 2003.

The statement is as follows:

Theorem. Let $F$ be a convex function, and let $\Omega\subset\mathbb{R}^n$ be bouded and open. Let $\varphi$ be Lipschitz continuous on $\partial\Omega$ with the Lipschitz constant $\leq k$. Then the functional $$ I(u)=\int_\Omega F(Du)\, dx $$ attains minimum in the class of $k$-Lipschitz functions on $\Omega$ that agrees with $\varphi$ on $\partial\Omega$.

Consider $n=1$, $A=[-1,1]$, $\Omega=(0,1)$. You want to minimize $\int_0^1 |u'(x)|\,dx$ subject to Dirichlet conditions, say $u(0)=0$ and $u(1)=1$. Then it is quite obvious that every monotone function is a minimizer, so uniqueness does not hold.