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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!

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  • $\begingroup$ Life span with Klainerman-Sobolev? What the hey? On a compact domain you have no dispersive decay and Klainerman-Sobolev absolutely cannot be applied. Besides, you are working with a linear inhomogeneous equation so as long as $F$ is sufficiently regular your classical solution exists for all time. On a compact domain you don't really want to commute with vector fields, since if the vector fields are transverse to the boundary it interchanges Dirichlet and Neumman boundary conditions. You would be better off commuting the Dirichlet Laplacian instead to get spatial regularity. $\endgroup$ Feb 18, 2015 at 9:42
  • $\begingroup$ Also, by what do you mean the boundary energy? By your assumptions only the normal derivative is non-vanishing on $\partial D$ so it seems a bit strange to write that as $\|u (t,\cdot)\|_{H^r(\partial D)}$, since $u|_{\mathbb{R}\times\partial D} \equiv 0$. $\endgroup$ Feb 18, 2015 at 9:55
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    $\begingroup$ On the other hand, you can get very crude estimates on $\| \partial_r u\|_{L^2([0,T]\times \partial D)}$ by using the multiplier field $r \partial_r = \sum x_i \partial_i$ and the fundamental energy estimate. To get higher regularity commute the equation by the angular momentum vector fields. It would be better if you can say something about what types of estimates are you looking for, or possibly what applications you are interested in. $\endgroup$ Feb 18, 2015 at 10:06
  • $\begingroup$ @WillieWong Thanks for your comments. You are definitely right, one cannot apply K-S on compact domains. What I really meant to say "if we are working in the $\mathbb{R}^{n}$, the we can conclude lifespan estimate via K-S." $\endgroup$
    – Yun K
    Feb 19, 2015 at 0:05
  • $\begingroup$ @WillieWong This is a problem arises from my current work on the free boundary problem of a fluid. What I'm really looking for is the connection between interior and boundary estimate. The inequality provided by Dr. Serre is exactly the type of inequality I'm looking for.In addition, I have tried to commute with rotational vector fields and apply classical energy estimate, since rotational fields preserves the boundary condition. However, I'm a bit worrying about the radial vector field, since it is in the normal direction of the boundary. How can I deal with that? $\endgroup$
    – Yun K
    Feb 19, 2015 at 0:29

1 Answer 1

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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.

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  • $\begingroup$ Thank you for the answer, Dr. Serre. This is exactly the type of energy estimate I'm looking for. In addition, I have checked out your book from my school library today. $\endgroup$
    – Yun K
    Feb 19, 2015 at 0:28

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