Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 105 / 1 / c / Solution

Choose $x$ at which $Du(x)$ is finite and the $p$-mean version of the Lebesgue differentiation theorem holds:These conditions hold almost everywhere. On a cube about $x$, subtract the affine function and put $v(y)=u^{\ast }(y)-u^{\ast }(x)-Du(x)\cdot (y-x)$. Its weak derivative is $Dv=Du-Du(x)$ and $v(x)=0$. Approximate $u$ as in part (b) and subtract the same affine function from the approximants. Uniform convergence and convergence of their weak derivatives extend part (a)'s estimate to $v$, givingFor $h\ne 0$, choose $r=4|h|$ and $y=x+h$. It follows thatThis is precisely classical Fréchet derivative differentiability, not merely existence of coordinate derivatives. Therefore the differentiability almost everywhere of supercritical Sobolev functions gives