The equation
\[ \Delta u = f, \]is called Poisson’s equation, where \(\Delta\) is the Laplace operator .
Remarks
- Laplace’s equation does not assume any boundary conditions .
Special cases Link to heading
- Dirichlet problem for Poisson’s equation
- Poisson’s equation on the whole space with compactly supported \(f\).
- A domain \(\Omega\), with a hole and a infinite metric tree in the hole, which is glued with the boundary of \(\Omega\), is covered in [1].
Solutions Link to heading
- If \(f \in C_c^2(\mathbb{R}^d)\) the convolution of the fundamental solution and \(f\), i.e. \[u(x) = \phi * f(x),\] solves Poisson’s equation. [2, 2.2 Theorem 1]
See also Link to heading
References Link to heading
- M. Kachanovska, K. Naderi, and K. Pankrashkin,
Poisson-type problems with transmission conditions at boundaries of infinite metric trees,
2025. doi:10.48550/arXiv.2506.11218 - L. Evans, Partial differential equations. Providence (R. I.): American mathematical society, 1998.