Poisson's equation

Remarks

Example: on $U = \R^n$ with $f = 1$, the function $u(x) = \abs{x}^2 / (2n)$ is a classical solution, since each term $\partial_i^2 (x_i^2 / (2n))$ equals $1/n$. Non-example: $u(x) = \abs{x}^2 / 2$ on $\R^2$ is not a solution for $f = 1$, since $\Delta u = 2$. With $f = 0$ the equation is Laplace's equation. If $u_1$ and $u_2$ solve the same Poisson equation, then $u_1 - u_2$ is harmonic. Many texts write the equation as $-\Delta u = f$, the sign that makes the operator positive. If $f$ is twice continuously differentiable with compact support in $\R^n$, $n \ge 2$, then $u = \Phi * f$ solves $-\Delta u = f$ on $\R^n$, where $\Phi$ is the fundamental solution of the Laplacian. Weak and distributional solutions are defined for less regular $f$.