The first eigenvalue of the Dirichlet- or Neumann-Laplacian on a bounded domain \(\Omega\) with suitable boundary regularity the minmax principle simplifies to
\[ \lambda_1 = \min_{\phi\in H^1(\Omega)} \frac{\lVert \nabla \phi\rVert^2_{L^2(\Omega)}}{\lVert \phi\rVert^2_{L^2(\Omega)}}. \]