Simple problems on Sobolev spaces

Here are some problems on Sobolev spaces.

Problem 1. Prove that if $\Omega \subset \mathbb{R}^n$ is connected, $u\in W^{1,p}(\Omega)$ and the weak derivatives

a.e. on $\Omega$, then $u\equiv\text{const}$ a.e.

Problem 2. Let $n>1$. Check that

belongs to $W^{1,n}(B(0,1/e))$, where $B(0,1/e)\subset\mathbb{R}^n$. Check that $\nabla u\not\in L^p$ for $p>n$.

Solve the next problem to understand that differentiability in the weak sense and differentiability a.e. in the classical sense are two different notions.

Problem 3. Let $v\colon[0,1]\to [0,1]$ be a continuous, nondecreasing function such $v(0)=0, v(1)=1$ and $v$ is constant on each segment in the complement of the standard Cantor set $C$. (I.e., the classical derivative $v'$ of $v$ exists and is equal to zero at points of $[0,1]\setminus C$.) Does $v$ have a weak derivative $v'\in L^1$?

Problem 4. Let $v\colon[0,1]\to \mathbb{R}$ be a piecewise $C^1$ function. Prove (directly by definition) that the weak derivative $v'\in L^1$ exists and is equal to the classical derivative $v'$ at points where $v$ is classically differentiable.

Problem 5. Let $u$ be locally integrable on $\mathbb{R}^n$. Assume that weak derivatives $D^\alpha u,D^\beta u$ exist. Prove that if two of the weak derivatives $D^\beta(D^\alpha u),\ D^\alpha(D^\beta u), D^{\alpha+\beta}u$ do exist, then all three exist and are equal. Can you weaken the assumptions?

Problem 6. Let $\Omega$ be a domain in $\mathbb{R}^n$. Assume that $u\in W^{1,p} (\Omega)$ for some $p\ge 1$. Prove that

Is the same true for $W^{2,p}$?

Hint: Consider $f_j\circ u$ for $f_j(t)=(t^2+j^{-2})^{1/2}-j^{-1}$, where $j=1,2,\ldots$.