= Lipschitz truncation preserves zero-boundary Sobolev spaces
{c}
{title2=$g(0)=0\ \Longrightarrow\ g(H_0^1)\subseteq H_0^1$}
If $v\in H_0^1(\Omega)$ and $g:\mathbb R\to\mathbb R$ is <Lipschitz continuous> with $g(0)=0$, then $g(v)\in H_0^1(\Omega)$. The Lipschitz version of the <Sobolev chain rule> controls its <weak derivatives>, and its boundary trace is $g(0)=0$. Equivalently, approximate $v$ by compactly supported smooth functions, apply the derivative bound to their compositions, and pass weakly in $H^1$ and strongly in $L^2$. The closed linear space $H_0^1$ is weakly closed. This allows positive-part and bounded truncations in weak formulations without losing the <Dirichlet boundary condition>.
Back to article page
