Uniform bounds for eigenfunctions of the Laplacian on manifolds with boundary

Let $u$ be an eigenfunction of the Laplacian on a compact manifold with boundary, with Dirichlet or Neumann boundary conditions, and let $-\lambda^2$ be the corresponding eigenvalue. We consider the problem of estimating the maximum of $u$ in terms of $\lambda$, for large $\lambda$, assuming $u$ is $L^2$-normalized. We prove that $\max_M u\leq C_M \lambda^{(n-1)/2}$, which is optimal for some $M$. Our proof simplifies some of the arguments used before for such problems. In order to make the article accessible to non-specialists, we review the 'wave equation method' (which has become standard in asymptotic eigenvalue problems) and discuss some special cases which may be handled by more direct methods.