Let \(U\subset \mathbb{R}^n\) be an open set and bounded in at least one direction. Then for \(1\le p<\infty\) exists a constant \(C\) depending on \(U\) and \(p\), such that for every \(u\in W^{1,p}_0(U)\) the Poincaré inequality holds

\begin{equation*} \lVert u\rVert_{L^p(U)}\le C\lVert Du\rVert_{L^p(U)}. \end{equation*}

Proof ideas Link to heading

Elementary proof Link to heading

Assume \(U\) is bounded in the \(e_n\) direction. Using the fundamental theorem estimate \((u(\widetilde{x}, x_n)\) with the derivative and then integrate over all variables. [1].

Using Sobolev embeddings where \(U\) is bounded Link to heading

Assume \(pSobolev conjugate \(p^*\) is larger then \(p\) we get the Poincaré inequality using (0x6742241f) and (0x6745ca7e) . For \(p\ge n\) choose a \(q< n\) with \(q^*>p\), then again using the Sobolev embedding and (0x6742241f) we obtain

\begin{equation*} \lVert u\rVert_{L^p(U)}\lesssim \lVert u\rVert_{L^q(U)}\lesssim \lVert Du\rVert_{L^{q^*}(U)}\lesssim \lVert Du\rVert_{L^p(U)}. \end{equation*}

References Link to heading

1. B. Schweizer, Partielle Differentialgleichungen: Eine anwendungsorientierte Einführung. Berlin, Heidelberg: Springer Berlin Heidelberg, 2023. doi:10.1007/978-3-662-67188-7