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Consider the uniformly elliptic equation $(\partial_t^2 + L)u = 0$ on $(0, \infty) \times \Omega$, where $\Omega \subset \mathbb{R}^n$ is an open bounded domain with smooth boundary, and $L$ is a second order negative definite elliptic operator (not necessarily of divergence type) with smooth coefficients. The domain of $L$ satisfies $\mathcal{D}(L) \subset H^2(\Omega),$ and is defined by the regular elliptic boundary conditions $B(x, \partial_x)u = 0$ on $\partial\Omega$. I am trying to find references to the following kinds of elliptic estimates: $$ \Vert Lu (1, .) \Vert_{L^\infty (\{1\} \times B\cap\Omega)}\leq C\Vert u (., .)\Vert_{L^2((\frac{1}{2}, \frac{3}{2}) \times B^{'}\cap\Omega)},$$ where $B \subset B^{'} \subset \mathbb{R}^n$ are open balls with centers inside $\Omega$. I would highly appreciate a reference. This is a bit similar to the following MO question.

Edit: It might suffice to obtain bounds like $\Vert u\Vert_{W^{k, 2}(I\times B\cap\Omega)} \lesssim \Vert u\Vert_{L^2(I^{'} \times B^{'}\cap\Omega)}$ for high enough $k$, where $I \subset I^{'}$ are open intervals in $(0, \infty)$, but I am not sure what happens at the boundary.

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  • $\begingroup$ Your question makes no sense. You start by asking for a solution to $Lu = 0$. Then $\|Lu\|_{L^\infty(B)} = 0$ a priori and so the kind of "elliptic estimate" you are asking about is trivially true by your set-up. Can you edit your question to be a bit more clear what you are after? $\endgroup$ – Willie Wong Aug 3 '16 at 21:30
  • $\begingroup$ @WillieWong Sorry about that, edited it now. $\endgroup$ – user95844 Aug 3 '16 at 21:41
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I am not an expert at this, but I will try to make some comments, with the hope that someone will correct me if I am wrong. The ellipticity and the boundary condition you consider together give you $\Vert u\Vert_{H^{k + 2}(\Omega)} \lesssim \Vert u\Vert_{L^2(\Omega)}$. This follows from Michael Taylor's PDE book, Vol I, Chapter 5, particularly equation (11.29). Now repeat this till you get $k$ high enough so that $H^k(\Omega) \hookrightarrow C^{2, \alpha}(\Omega)$. Then uniform ellipticity on $L$ will give you something like $\Vert Lu\Vert_{L^\infty(\Omega)} \lesssim \Vert u\Vert_{L^2(\Omega)}$. This is a global estimate, to convert it into the one you want, I am guessing that you will need to insert cut-off functions.

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