Let $ \Omega $ be a smooth bounded domain in $ \mathbb{R}^d $ and $ T>0 $ be a positive number. Consider the wave equation in the domain $ \Omega\times(0,T) $ \begin{align} \left\{\begin{matrix} \partial_t^2u\Delta u=F&\text{ in }&\Omega\times(0,T),\\ u=0&\text{ on }&\partial\Omega\times(0,T),\\ u(x,0)=f(x),\partial_t u(x,0)=g(x),&\text{ on }&\Omega\times\left\{0\right\}. \end{matrix}\right. \end{align} It is well known that for $ f\in H_0^1(\Omega) $, $ g\in L^2(\Omega) $ and $ F\in L^2(\Omega\times(0,T)) $, we can construct a unique weak solution $ u\in W^{2,2}(0,T;H^{1}(\Omega))\cap W^{1,\infty}(0,T;L^2(\Omega))\cap L^{\infty}(0,T;H_0^1(\Omega)) $. I am interested in the a priori estimates for $ u $. I have already know that for $ \Omega=\mathbb{R}^d $, we can get Strichartz estimates. I wonder if there is similar Strichatz estimates for the wave equation with $ \Omega $ is bounded. Moreover, if we change $ \Delta $ to $ \operatorname{div}(A(x)\nabla) $ what is the result? Can you give me some references or hints?

$\begingroup$ If you are content with just energy estimates instead of Strichartz estimates (the difficulty in obtaining which was pointed out in detail by Willie Wong's answer), these do exist for mixed DirichletCauchy problems such as yours. See e.g. Volume III of L. Hörmander, "The Analysis of Linear Partial Differential Operators", more precisely Section 24.1. These are used e.g. to establish the results you quoted. On the other hand, any dispersive / decay information you may want is of course lost in such estimates. $\endgroup$– Pedro Lauridsen RibeiroMay 7, 2022 at 5:55
1 Answer
Strichartz estimates on domains is a difficult problem!
First: on bounded domains you cannot have any global in time Strichartz estimates. This is because of the presence of standing waves. (Set initial data to be an eigenfunction of the Laplacian.)
On the other hand, there is still the possibility of local in time Strichartz. But recall that Strichartz estimates capture dispersive phenomenon, where two wave packets starting out at the same location with different velocities will separate spatially. When you work on a domain, the wave packet may now hit the boundary and reflect back. So you will expect some degree of losses due to such "singularities".
There is a lot of research on how to understand this. I would suggest starting by looking up papers by Oana Ivanovici and tracing through the literature.
(The problem on "exterior" domains are somewhat easier, especially when the domains have nice boundaries such that each wave packet can only hit it at most once and reflect. [For example, when the domains are convex.] In those cases Strichartz estimates have been proven.)
Finally: Strichartz estimates for variable coefficient wave equations (on the whole space) have also been previously studied. Look up papers by Hart Smith and Daniel Tataru (together and separately).

1$\begingroup$ I believe the 1993 JAMS paper by Mockenhaupt, Seeger and Sogge was the first to establish the (spacetime) local Strichartz estimates for varyingcoefficient wave and KleinGordon equations in full generality. These are also discussed in the last chapter of the book "Fourier Integrals in Classical Analysis" by the last author. $\endgroup$ May 6, 2022 at 2:00

$\begingroup$ @PedroLauridsenRibeiro Thanks for the addition! I have not looked deeply at that case so my knowledge of the literature is definitely spotty. My answer above is intended to give the OP some places to start looking, and is absolutely not meant to be exhaustive. $\endgroup$ May 6, 2022 at 13:10