Let \( U \subset \mathbb{R}^{d-1} \) be an open set and \( K = U \times (0,h) \) for \( h > 0 \).

Then, for \( u \in C^{1}(\overline{U}) \),

\[ \|u\|_{L^{2}(U)} \le C \|u\|_{H^{1}(K)}, \]

with

\[ C = \left( \frac{2}{h} + 1 \right)^{1/2}. \]
Remarks
  • The proof is based on the first step of the proof of the trace theorem [1, 5.5 Theorem 1].
  • It is not difficult to extend the result on \(1\le p< \infty\). In this case, \(C\) become dependent on \(p\).
Proof

Let \( \eta \in C^{\infty}(\mathbb{R}) \) be a smooth transition function, with

\[ \eta(y) = \begin{cases} 0, & y \ge h, \\ 1, & y \le 0. \end{cases} \]

Then

\begin{align*} \int_{U} |u(x',0)|^{2} \, dx' &= \int_{U} \eta(0) |u(x',0)|^{2} \, dx'\\ &= - \int_{U} \partial_{d} (\eta |u|^{2}) \, dx \\ &= - \int_{U} \eta' |u|^{2} + 2\eta u \, \partial_{d} u \, dx\\ &\le \left( \frac{2}{h} + 1 \right) \int_{K} \big( |u|^{2} + |\nabla u|^{2} \big) \, dx, \end{align*}

where we employed Cauchy’s inequality .

References Link to heading

  1. L. Evans, Partial differential equations. Providence (R. I.): American mathematical society, 1998.