The \(m\)-th eigenvalue of the Dirichlet- or Neumann-Laplacian on a bounded domain \(\Omega\) with suitable boundary regularity is given by

\[ \lambda_m = \min \max \frac{\lVert \nabla v\rVert^2_{L_2(\Omega)}}{\lVert v\rVert^2_{L^2(\Omega)}} \]

where the maximum is taken over all linear combinations of the form

\[ v = a_1\phi_1 + \cdots + a_m \phi_m \]

and the minimum is over all choices of \(m\) linearly independent functions \(\phi_1,\ldots ,\phi_m\in H^1(\Omega)\) (see [1] and the references therein).

Links Link to heading

• Laplacian
• first eigenvalue

References Link to heading

1. D. Grebenkov and B. Nguyen, Geometrical Structure of Laplacian Eigenfunctions, SIAM Review, vol. 55, no. 4, p. 601–667, 2013. doi:10.1137/120880173