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I have seen some regularity result for ellptic PDE but all of them consist of uniform elliptic one. For instance, $$\nabla \cdot (\gamma(x) \nabla u)=F \text{ in } \Omega\qquad u=\phi \text{ on }\partial \Omega$$ with $0<\lambda\le \gamma(x)\le \Lambda<\infty$. Regularity result I am interested are $$\|u\|_{2,\Omega}^2\le C(\|F\|_{0,\Omega}+\|\phi\|_{1/2,\partial\Omega})$$ Is there any result available if we have condition $\gamma(x)\ge 0$?

Any hint or reference will be greatly appreciated.

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    $\begingroup$ The equation you are dealing with in not exactly a elliptic equation, but it is an equation with non positive characteristics: perhaps the references cited in this answer may contain something useful for you. $\endgroup$ Aug 16, 2021 at 16:36

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