Does this time-dependent trace space have a name?

Question

This question is a follow up to this question.

Let $\Omega \subset \mathbb{R}^d$ be an open connected set. For each $t\in \mathbb{R}^+$ let $u_d:\partial\Omega \to \mathbb{R}$ be in $H^{1/2}(\partial \Omega)$. Let $u_e : \Omega \to \mathbb{R}$ be an extension of $u_d$ from the boundary to the interior.

I am interested in the set of all $u_d$ for which we can find an extension $u_e$ such that

$$ \left\| u_e\right\|_{L^2(\Omega)} + \left\| \nabla u_e\right\|_{L^2(\Omega)} + \left\|\frac{\mathrm{d}}{\mathrm{d}t} u_e\right\|_{L^2(\Omega)} $$

is finite for all $t\in \mathbb{R}^+$. What restrictions do I have to place on $u_d$ to have a well-defined time derivative of the extension? In addition, I am looking for a reference and a name for this space.

Answer 1

If you insist on "for all $t$ " as distinct from "for almost every $t$ ", one possible space is $C^1([0,\infty);H^{-1/2})\cap C^0([0,\infty);H^{1/2})$. A possible extension is the unique harmonic function in $\Omega$, the minimizer of $\int_\Omega |\nabla v|^2$ : this extension operator $E$ is well known to map $H^{1/2}(\partial\Omega)$ to $H^1(\Omega)$ but it also satisfies $||Eu||_{L^2}\leq||u||_{H^{-1/2}}$, as you can see with its expression in the case of the upper half-space: for $d=2$,$$Eu(x,y)=\frac{y}{\pi }\int_\mathbb{R} \frac{u(x'-x)\ dx'}{x'^2+y^2}$$so that $\mathcal{F} Eu(.,y)(\xi)=e^{-y|\xi|}\mathcal{F}u(\xi)$ (then use $\int e^{-2y|\xi|^2}\ dy=1/2|\xi|^2$ to bound $\int Eu(x,y)^2\ dx\ dy$) and mutatis mutandis for $d\ge2$ and more general (smooth enough) $\Omega$.

If you're content with "for almost every $t$ and as distributions", then replace $C^1$ with $W^{1,1}$ and $C^0$ with $L^1$ (if no bound uniform in $t$ is needed) or with $W^{1,\infty}$ and $L^\infty$ if it is.