A Trace Theorem for Solutions of Linear Partial Differential Equations
Symes, William W.
The main goal of this work is to determine circumstances under which the trace of the solution of a linear partial differential equation is as smooth as the solution itself. Clearly, if the linear partial differential equation is strictly hyperbolic with smooth coefficients, standard energy estimates will yield the fact that the solution along any spacelike trace is as smooth as itself locally, provided a sufficiently smooth right-hand side. Unfortunately, for more general equations or even a strictly hyperbolic differential equation but this time along a nonspacelike trace, the same idea will not work, essentially because one does not know how to apply energy estimates to a nonhyperbolic problem directly. In this paper, we shall investigate the trace regularities of solutions to linear P.D.E. Our result shows that the difficulties discussed above may be cured by imposing some additional microlocal smoothness.
Citable link to this pagehttps://hdl.handle.net/1911/101664
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