Suppose we have the wave type equation $$\partial^2_tu - L u = 0$$ on a compact manifold with boundary, where $L$ is a second order strongly elliptic operator with coercive boundary conditions (not necessarily Dirichlet or Neumann) making it self-adjoint. I wanted some references for $L^p$ regularity results for such equations, if such results exist at all. I don't think the standard Calderon-Zygmund theory or the standard pseudodifferential approach hold because of the boundary conditions. Thank you in advance.
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$\begingroup$ I don't see how you could expect elliptic regularity for a hyperbolic equation.. $\endgroup$– Matthias LudewigCommented Jul 12, 2014 at 9:28
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$\begingroup$ @Kofi Oops, my bad! Corrected. $\endgroup$– GuestCommented Jul 12, 2014 at 11:23
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$\begingroup$ Can you specify your boundary condition, and your assumptions on the initial data? $\endgroup$– Mark PeletierCommented Jul 17, 2014 at 5:12
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