Let $D$ be the unit disk in $\mathbb{R}^{n}$, we consider the $n$ dimension wave equation defined on $D$,
$$\square u=F$$

where $\square=\partial_{t}^{2}-\triangle$ is the standard wave operator in $\mathbb{R}^{1+n}$, together with the initial and boundary conditions
$$u(0,x)=u_{0},u_{t}(0,x)=u_{1}, u(t,x)|_{\partial D}=0$$
A well-known theorem states that
$$||\partial u(t,x)||_{L^{2}(D)}\leq ||\partial u(0,x)||_{L^{2}(D)}+\int_{0}^{t}||F||_{L^{2}(D)}$$
In addition, one can apply the above energy inequality and Sobolev lemmas to estimate the higher order energy by commuting $\square$ with vector fields.

**My question is: Can we apply a similar method and get the higher order energy estimate on the boundary $\partial D$?**

My ideas are as follows:

We pick $\mu_{i}$ and $\nu_{i}$ to be radial functions which form a partition of unity subordinate to the sets $\{x\in\mathbb{R}^{n}:|x|\leq 1-\frac{d}{2i}\}$ and $\{x\in\mathbb{R}^{n}:|x|\geq 1-\frac{d}{i}\}$, respectively, where $d$ is chosen such that the normal $N$ to $\partial D$ can be extended to $\{x\in\mathbb{R}^{n}:|x|\geq 1-d\}$.By doing that, the boundary energy $||u(t,\cdot)||_{H^{r}(\partial D)}$ can then be approximated by estimating $||\nu_{i}u(t,x)||_{H^{r}(D)}$. But I don't know if I can take the limit when $i\to \infty$.

Use of the extension operator $E$ to extend $u$ as a compactly supported function defined in $\mathbb{R}^{n}$. But I don't know what to do for the next...

So I was wondering if anyone can help me out at this point? I'm always open to new ideas!

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