Suppose $\Omega$ is a bounded domain in $\mathbb{R}^n.$ Let $w\in H^{1,2}$ (standard Sobolev space, order 1, integrability 2) and $L>0$ be given. Is it then true that the function $w\_L:=\min (L,w)$ is also in $H^{1,2}?$


I found this assertion in the book "Riemannian geometry and geometric analysis" of Jost, in the section concerning higher regularity of harmonic maps. There, one already knows that $f$ (the continuous weakly harmonic map) is in $H\_{loc}^{1,4} \cap H^{2,2}\_{loc}$ and considers $w:=|Df|^2.$


Now, the function $w\_L$ can also be written as $x\mapsto w(x)\chi\_L(w(x)),$ where $\chi\_L$ denotes the characteristic function of the set consisting of all x s.t. $x\le L$. Considering the distributional derivative of $w\_L$ gives you something involving a delta function(al), which is not in $L^2.$


Any suggestions? What am I doing wrong?



cutting off $H^{1,2}$-functions in the image
Note that $\min(w,L) = \frac{1}{2} (w+L) - \frac{1}{2} |w-L|$, so the main issue is to establish that the map $w \mapsto |w|$ is bounded on $H^{1,2}$. But this follows from the diamagnetic inequality $|\nabla |w|| \leq |\nabla w|$ (in the sense of distributions), which is obvious formally, but can be established rigorously by approximating the absolute value $|x|$ by the smoothed variant $(|x|^2+\varepsilon^2)^{1/2}$ and then letting $\varepsilon$ go to zero. (The diamagnetic inequality can be found for instance in the text of Lieb and Loss; it is more commonly applied in the context of covariant differentiation, but already has usefully non-trivial content for ordinary differentiation.)


More generally, composition with Lipschitz functions will preserve all $W^{s,p}(\Omega)$ spaces for $0 \leq s \leq 1$ and $1 < p < \infty$ by the chain rule (if $s=1$) or fractional chain rule (if $s<1$), using regularisation arguments as necessary to make the argument rigorous. (See for instance Taylor's book "Tools for PDE" for this sort of thing.)


