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Specifically, I have $Lu=f$, where $L$ is a linear elliptic pseudodifferential operator, on an unbounded domain of the form $\Omega\times [0,\infty)$, $\Omega$ has Lipschitz boundary, $u$ is 0 on all boundaries, i.e. $(\partial\Omega\times[0,\infty))$ $\cup$ $(\Omega\times\{0\})$. I imagine that there are existence theorems that contain this problem as a special case, but the results I have found deal with bounded domains.

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There are existence theorems, but they require more information and conditions than you give. Any chance you want to say more about what you want or need? –  Deane Yang May 3 '11 at 3:51
This is the basic framework, but I want to leave the problem vague enough to accomodate any additional conditions that an existence theorem might require. I am flexible :) –  Max May 3 '11 at 17:52
Hi Max. To get Deane's attention again you should ping him with @Deane. By the way, which books have you looked through? This seems very standard. It should be in Hörmander ;). –  Glen Wheeler May 16 '11 at 9:54
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