Ritz-Rayleigh quotient and Laplace Eigenfunction
Consider the eigenvalue problem of the Laplace operator on \(\Omega\subset\mathbb{R}^n\) with Dirichlet boundary condition: \[ \begin{cases} -\Delta u = \lambda u & x\in\Omega \\ u = 0 & x\in\partial\Omega \end{cases}. \]
We'll begin by illustrating examples when \(n=2\), where \(\Omega\) represents commonly encountered shapes. In these scenarios, we'll use \(x\) and \(y\) to denote the input variables of \(u\) instead of \(x\in\mathbb{R}^2\). The variational formulation of the solution through Ritz-Rayleigh quotient is \[ \lambda = \inf_{u\in H_{0}^{1}(\Omega)} \frac{\int_{\Omega} |\nabla u|^2 \mathrm{d} x}{\int_{\Omega} u^2 \mathrm{d} x}, \] where the infimum is attainable, and the minimizer is the solution to the Dirichlet BVP of the Laplace operator associated with the smallest eigenvalue \(\lambda_1(\Omega)\).