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:


1. 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 $i\to \infty$.
2. 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!



Boundary energy estimate of wave equations
This is a classical but intricate question : what are the boundary and interior estimates for a hyperbolic initial-boundary value problem ? The answer is clean for the wave equation, because the boundary is non-characteristic. You should consider $L^2$-spaces in both space and time variables. A typical estimate is
$$\gamma\int\_0^T\int\_De^{\gamma t}|\partial u|^2dxdt+\int\_0^T\int\_{\partial D}e^{\gamma t}|\partial u|^2ds(x)dt\le C\left(\int\_De^{\gamma t}|\partial u(0,x)|^2dx+\frac1\gamma\int\_0^T\int\_De^{\gamma t}|F|^2dxdt\right)$$
for every $\gamma,T>0$. You can use this inequality to estimates higher order norms.


Reference are a book by Chazarain & Piriou (1981) and mine (co-authored with S. Benzoni-Gavage) *Multi-dimensional hyperbolic partial differential equations. First order systems and applications.* Oxford University Press (2007). Mind that our book deals with first-order systems, but it applies to the wave equation by rewriting it as
$$v\_t-{\rm div}z=F,\qquad z\_t-\nabla y=0.$$


By the way, *non-characteristic* means that the wave operator is second-order in the direction normal to the boundary.


