Fixing a variable of a $H^1_0(\Omega)$ function

Question

Let $\Omega \subset \mathbb{R}^{N} = \mathbb{R}^{N_1} \times \mathbb{R}^{N_2}$. Suppose that $\Omega \subset A_1 \times A_2$, where $A_i \subset \mathbb{R}^{N_i}$. If $u \in C^\infty_0(\Omega)$, I know that, for a fixed $x \in A_1$, the function $v_x(y) := u(x,y)$ is in $C^{\infty}_0(A_2)$. I wonder if the same happens for a function in $H^1_0(\Omega)$, that is, if $u \in H^1(\Omega)$, then $v_x \in H^1_0(A_2)$?

Answer 1

This is a particular case of Sobolev embedding theorems. See Adams, Sobolev spaces, Theorem 5.4. There, embedding theorems are formulated in a more general way including your situation. That is embedding theorems are given for $W^{m,p}(\Omega) \hookrightarrow W^{j,q}(\Omega^k)$, where $\Omega^k$ is the intersection of $\Omega$ and a $k$-dimensional hyperplane.