Green's Function for Poisson's Equation
Summary
In this lecture we let $U\subset \mathbb{R}^n$ be open and bounded with $C^1$ boundary $\partial U$, and obtain a representation for solutions $u\in C^2(\overline{U})$ of the boundary-value problem $$ \left\{ \begin{alignedat}{2} -\Delta u &= f\quad &&\text{in}~U\\ u&=g\quad&&\text{on}~\partial U \end{alignedat}\right. $$ by introducing Green’s functions.
Full Set of Lecture Notes
The notes for this lecture are available here (21 pages).