Laplace’s equation is given by
\[ \Delta u = 0, \]where \(\Delta\) is the Laplace operator .
Remarks
- A solution of this equation is called harmonic function .
- The Laplace’s equation is radial-symmetric, i.e. if \(u\) is a solution, then \(v=u\circ O\) is also a solution for some orthogonal matrix \(O\).
- Laplace’s equation does not assume any boundary conditions
. But for the well posedness we need some if the domain is bounded. There are different options:
- Dirichlet boundary conditions
- Neumann boundary conditions
- Robin boundary conditions