The Poincare inequality for zero-boundary Sobolev closures on domains bounded in one direction

The smooth case: pointwise bound. Let $\phi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega };\mathbb{K})$ and extend it by zero to ${\mathbb{R}}^n$. Write $x=x_{\perp }+x_{n}e$ with $x_{\perp }\perp e$. For each fixed $x_{\perp }$, the profile $g(s):=\phi (x_{\perp }+se)$ is smooth and supported in $(a,b)$, since $\mathrm{\Omega }\subset \{a<y\cdot e<b\}$. If $x=x_{\perp }+x_{n}e\in \mathrm{\Omega }$, then $a<x_n<b$ and $g(a)=0$. On $[a,x_n]$, each real component of $g^{\mathrm{\prime }}$ is continuous, so [F8] supplies its Riemann integrability and converts its Riemann integral to its Lebesgue integral. Thus [F2] applied componentwise and the complex integral convention in [F10] give $\phi (x)=g(x_n)-g(a)={\int }_a^{x_n}D\phi (x_{\perp }+se)\cdot eds$. The continuous derivative and its modulus are Lebesgue integrable on this compact interval by [F8]; [F10] and $\mathrm{\mid }D\phi \cdot e\mathrm{\mid }\le \mathrm{\mid }D\phi \mathrm{\mid }$ therefore give $\mathrm{\mid }\phi (x)\mathrm{\mid }\le {\int }_a^{x_n}\mathrm{\mid }D\phi (x_{\perp }+se)\mathrm{\mid }ds$. For $p=1$ the resulting integral bound is already the desired estimate; for $p>1$, Holder [F3] applied to the real nonnegative function $\mathrm{\mid }D\phi \mathrm{\mid }$ and the constant $1$, followed by $x_n-a\le b-a$ and enlargement of the integration interval, yields $\mathrm{\mid }\phi (x){\mathrm{\mid }}^p\le (b-a{)}^{p-1}{\int }_a^b\mathrm{\mid }D\phi (x_{\perp }+se){\mathrm{\mid }}^{p}ds$.

The smooth case: integration. If $n=1$, then $\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }\le b-a$, so integrating step 1.1 gives ${\int }_{\mathrm{\Omega }}\mathrm{\mid }\phi {\mathrm{\mid }}^p\le (b-a{)}^p{\int }_a^b\mathrm{\mid }D\phi (se){\mathrm{\mid }}^{p}ds=(b-a{)}^p{\int }_{\mathrm{\Omega }}\mathrm{\mid }D\phi {\mathrm{\mid }}^p$ by [F9] for the global linear diffeomorphism $s\mapsto se$ of $\mathbb{R}$ (where $e=1$ or $e=-1$ and the absolute Jacobian is $1$), and the support of $D\phi$ in $\mathrm{\Omega }$. Now suppose $n\ge 2$. Choose orthonormal coordinates with last vector $e$ and write $x=x_{\perp }+x_{n}e$; the corresponding orthogonal map $T:{\mathbb{R}}^{n-1}\times \mathbb{R}\to {\mathbb{R}}^n$ is a global $C^1$ diffeomorphism with $\mathrm{\mid }\det DT\mathrm{\mid }=1$. By [F9], nonnegative integrands have the same integral after composing with $T$; this applies both to ${\mathbf{1}}_{\mathrm{\Omega }}\mathrm{\mid }\phi {\mathrm{\mid }}^p$ and to $\mathrm{\mid }D\phi {\mathrm{\mid }}^p$, which are Borel measurable since the zero extension is smooth and $\mathrm{\Omega }$ is open. For each $x_{\perp }$, the slice $S_{x_{\perp }}:=\{x_n:(x_{\perp },x_n)\in \mathrm{\Omega }\}$ is a measurable subset of $(a,b)$, so $\mathrm{\mid }{S}_{x_{\perp }}\mathrm{\mid }\le b-a$. Integrating step 1.1 over $\mathrm{\Omega }$ and applying Tonelli [F4] gives ${\int }_{\mathrm{\Omega }}\mathrm{\mid }\phi {\mathrm{\mid }}^{p}dx\le (b-a{)}^{p-1}{\int }_{{\mathbb{R}}^{n-1}}{\int }_{S_{x_{\perp }}}{\int }_a^b\mathrm{\mid }D\phi (x_{\perp }+se){\mathrm{\mid }}^{p}dsd{x}_{n}d{x}_{\perp }\le (b-a{)}^p{\int }_{{\mathbb{R}}^{n-1}}{\int }_a^b\mathrm{\mid }D\phi (x_{\perp }+se){\mathrm{\mid }}^{p}dsd{x}_{\perp }$. The last integral equals ${\int }_{{\mathbb{R}}^n}\mathrm{\mid }D\phi {\mathrm{\mid }}^{p}dx={\int }_{\mathrm{\Omega }}\mathrm{\mid }D\phi {\mathrm{\mid }}^p$, since $\phi$ and its gradient vanish outside $\mathrm{\Omega }$ and $\mathrm{\Omega }$ lies in the slab. Therefore $\mathrm{\parallel }\phi {\mathrm{\parallel }}_{L^p(\mathrm{\Omega })}^p\le (b-a{)}^p\mathrm{\parallel }D\phi {\mathrm{\parallel }}_{L^p(\mathrm{\Omega })}^p$.

The general class. By [F1] there are ${\phi }_k\in C_c^{\mathrm{\infty }}(\mathrm{\Omega };\mathbb{K})$ with ${\phi }_k\to u$ in $W^{1,p}(\mathrm{\Omega };\mathbb{K})$; by step 2.1, $\mathrm{\parallel }{\phi }_k{\mathrm{\parallel }}_{L^p}\le (b-a)\mathrm{\parallel }D{\phi }_k{\mathrm{\parallel }}_{L^p}$ for every $k$. Both sides are continuous in the $W^{1,p}$ norm: $\mathrm{\parallel }{\phi }_k{\mathrm{\parallel }}_p\to \mathrm{\parallel }u{\mathrm{\parallel }}_p$ and $\mathrm{\parallel }D{\phi }_k{\mathrm{\parallel }}_p\to \mathrm{\parallel }Du{\mathrm{\parallel }}_p$ by [F6]. Passing to the limit gives $\mathrm{\parallel }u{\mathrm{\parallel }}_{L^p(\mathrm{\Omega })}\le (b-a)\mathrm{\parallel }Du{\mathrm{\parallel }}_{L^p(\mathrm{\Omega })}$, so the asserted inequality holds with $C(p)=1$.