Poisson's Equation
Summary
This lecture considers Poisson’s equation $$ -\Delta u = f\quad \text{in}~ \mathbb{R}^n, $$ where $f\colon \mathbb{R}^n \to \mathbb{R}$ is a given function. We cover some essential identities and theorems from calculus such as Gauss’ (Divergence) Theorem and Green’s identities. We prove the following. Let $$ u(x):= \int\limits_{\mathbb{R}^n} \Phi(x-y) f(y)\,\mathrm{d}y, $$ where $\Phi$ is the fundamental solution of Laplace’s equation constructed in Lecture 3 and $f\in C^2_\mathrm{c}(\mathbb{R}^n)$. Then
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$u\in C^2(\mathbb{R}^n)$.
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$-\Delta u = f$ in $\mathbb{R}^n$