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I asked a similar question the other day, but I will be more precise now.
Consider $ \Omega:=(0,1 ) \times (0,1)$ and consider
$$ - u_{xx}(x,y) - u_{yy}(x,y) + a(x) u_x + b(y) u_y + u = f(x,y) \mbox{ in } \Omega $$ with $ \partial_\nu u=0$ on $\partial \Omega$ (assume one has some sort of variational solution $u \in H^1(\Omega)$. Suppose $ a(x),b(y)$ are smooth in $ \Omega$ and assume $f$ is Holder continuous on $ \Omega$. \

By looking at the even extension of $u$ (across the various boundaries and which then makes one look at odd extensions of $a$ and $b$ which have jump discontinuities) i seem to be able to prove that $ u \in C^{2,\alpha}( \overline{\Omega})$. I am curious whether this result is true or ? So my question is: does this regularity seem correct or is it known one does not have this regularity at the corners? (the result suprised me a bit but I am also not familiar with what to expect with Neumann boundary conditions).
thanks a bunch for any comments.

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With appropriate regularity conditions, this is the conclusion of Theorem 6.4.2.6 in:

Grisvard, P.(F-NICE) Elliptic problems in nonsmooth domains. Monographs and Studies in Mathematics, 24.

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