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Is there a $W^{2,p}$ estimate or $C^{2,\alpha}$ estimate for Poisson equations in the whole space $\mathbb{R}^n$? That is, suppose $f\in L^{p}(\mathbb{R}^n),1<p\leq\infty(\in C^{\alpha})$, and $\Delta u=f \text{ in }\mathbb{R}^n$. Do we have $u\in W^{2,p}(\mathbb{R}^n)(\in C^{2,\alpha})$ ? And for more general elliptic equations? Any counterexamples or reference?

Thanks!

Ray Yang
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Y.Z
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1 Answers1

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Yes. You should look in Gilbarg-Trudinger. Chapter 9 does the $L^p$ estimates, and I think Chapter 4 does the Holder regularity estimates for the Poisson equation - chapter 6 does the more general Schauder estimates.

Ray Yang
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