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 −λ2 be the corresponding eigenvalue. We consider the problem of estimating maxM u in terms of λ, for large λ, assuming ∫ M u 2 = 1. We prove that maxM u ≤ CMλ (n−1)/2, which is optimal for some M . Our proof simplifies some of the arguments used before for such problems. We review the ’wave equation method’ and discuss some special cases which may be handled by more direct methods.