We call a function \(u\colon \Omega\to \mathbb{C}\) an eigenfunction of the Laplacian if there is \(\lambda\in \mathbb{C}\) such that

\begin{equation*} -\Delta u = \lambda u. \end{equation*}

The number \(\lambda\) is called eigenvalue.

Links Link to heading

• Laplacian
• eigenfunctions of the Laplace-Beltrami operator

Questions Link to heading

• For which \(\Omega\) eigenvalues exist and why?
• What kind of regularity the eigenfunctions have?