Energy Methods for Poisson's Equation
Summary
In this lecture we cover energy methods for the boundary-value problem (BVP) $$ \left\{ \begin{alignedat}{2} -\Delta u &= f\quad &&\text{in}~U\\ u&=g\quad&&\text{on}~\partial U \end{alignedat} \right. $$ We define the energy functional $$ I[w]:= \int\limits_U \frac{1}{2} |D w(y)|^2 - w(y)f(y) \, \mathrm{d}y $$ for $w$ belonging to the admissible set $$ \mathcal{A}:= \left\{w\in C^2(\overline{U})~\vert~ w=g~\text{on}~\partial U\right\}, $$ and among other things, cover Dirichlet’s Principle:
$u\in C^2(\overline{U})$ solves (BVP) if and only if $I[u]=\min_{w\in\mathcal{A}} I[w]$.
Full Set of Lecture Notes
The notes for this lecture are available here (8 pages).