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I am interested in the regularity of ellitpic equations like

$$ -\Delta u(x) +a(x) \cdot \nabla u(x) + C(x) u(x) =f(x) \quad \Omega$$ with $ \partial_\nu u =0$ on $ \partial \Omega$ where $ \Omega=(0,1) \times (0,1)$.

Assuming $a,C$ and $f$ are smooth, how much regularity can one expect for $u$. Lets assume apriori we know $u$ is bounded and say in $H^1(\Omega)$ (say its a variational solution). How about if we weaken $ f \in L^p$ for $1<p<\infty$. What are the $W^{2,p}$ (and more importantly) what Holder spaces can i expect $u$ to be in. \

thanks

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  • $\begingroup$ Just realized that since its a square and since we have zero Neumann, we can try and extend evenly across the boundary and solve on large domain... then use interior regularity theory??? is this the idea ? thanks $\endgroup$
    – Math604
    Commented Sep 24, 2015 at 21:09

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