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I know that let $f\in C_c^{2}(\mathbb{R}^n)$ and $u=\phi*f$ where $\phi$ is the fundamental solution of $-\Delta$, then $u\in C^2(\mathbb{R}^n)$ and $u$ solves the Poisson equation $-\Delta u=f$ in $\mathbb{R}^n$.

My question is, does this theorem hold good when $f$ is taken from some larger space of functions in $\mathbb{R}^n$ ? For example, $C^1(\mathbb{R}^n)$, $ C^\alpha(\mathbb{R}^n)(0<\alpha<1)$ and $ C^0(\mathbb{R}^n)$...

What is the best result up to date? Is there some references about this problem?

joriki
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Jianxing
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  • You can start your search in Gilbarg-Trudinger. – Mr. Brown Feb 24 '23 at 15:47
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    You can solve the Poisson equation in a distributional sense for any distribution $f$ such that the convolution $f*G$, where $G$ is the fundamental solution, makes sense. For instance, when $f$ is a distribution of compact support or when $f\in L^1$. If you want $u\in C^2$, you need $f\in C^{\alpha}$, in which case $u\in C^{2+\alpha}$, but $f\in C^0 \implies u\in C^2$ is false in general. For some basic results you can try Folland's PDE book, for more in-depth results you can look in Gilbarg-Trudinger's Elliptic PDE book. – kieransquared Feb 24 '23 at 16:57
  • Possible duplicate: https://math.stackexchange.com/q/3617484/532409 – Quillo Feb 22 '24 at 14:22

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