5

Evan's PDE book discusses Poisson's Equation $-\Delta u= f$ where $f\in C_c^2(U)$ as Theorem 1.1. With such a condition on $f$, we can basically pass all differentiations to it in order to show that $u\in C^2$. However, if we require only that $f \in C_c^1(U)$ then it seems that this method doesn't work so easily. I've attempted to do it by integration by parts, but it doesn't seem to work.

Following Evans, we can get $u_{x_i}=\int_{\mathbb{R}}\Phi(y) f_{x_i}(x-y) dy$ just as before. This shows $u$ is $C^1$. However, we can't put a second partial on $f$, so we must pass the derivative to $\Phi$ somehow, but this is where I am struggling since we need to avoid the blow up of $\Phi$ at $0$.

user3094803
  • 51

1 Answers1

2

This is a very good question, and the solution is actually might be much harder then you expect. Actually you could even relax the condition of $f$ to be bounded and locally Holder continuous (with exponent $\alpha\leq 1$) and you will have $u\in C^2(\Omega)$ and $\Delta u=f$ in $\Omega$.

I am not going to write the solution complete here but just refer to where you could find it. Please look at book by Gilbarg & Trudinger, page 54-56, lemma 4.1 and lemma 4.2. It is not a short prove.

spatially
  • 5,472