Boundary energy estimate of wave equations

Question

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!

Answer 1

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.