For \(d \geq 2\), \(\phi: \mathbb{R}^d \to \mathbb{R}\)
\[\phi(x) = \begin{cases} -\frac{1}{2\pi} \log |x|, & d = 2 \\ \frac{1}{d(d-2)\omega_d} \cdot \frac{1}{|x|^{d-2}}, & d \geq 3 \end{cases}\]is the fundamental solution of the Laplace’s equation , where \(\omega_d\) is the volume of the unit ball.
Remarks
- The fundamental solution is harmonic in \(\mathbb{R}^d\setminus \{0\}\) and has a singularity in \(0\).
- One way to find the fundamental solution is described in [1, 2.2]. Since the Laplace’s equation \(\Delta\) is radial-symmetric, search for solutions of the type \(u(x) = u(|x|)\). Then \[\Delta u = u' + \frac{d-1}{r} u'' = 0.\] and we essentially obtain for the first derivative \[\implies u' = \frac{c}{r^{d-1}}.\] Integration leads to the fundamental solution.
See also Link to heading
- Solution of the Poissons equation on \(\mathbb{R}^d\) and compactly supported \(f\).
- Greens function
References Link to heading
- L. Evans, Partial differential equations. Providence (R. I.): American mathematical society, 1998.